微积分学示例

$\int \cos^{6} (\frac{y}{2}) d y$

解题步骤 1

使 $u_{1} = \frac{y}{2}$ 。然后使 $d u_{1} = \frac{1}{2} d y$ ，以便 $2 d u_{1} = d y$ 。使用 $u_{1}$ 和 $d$ $u_{1}$ 进行重写。

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解题步骤 1.1

设 $u_{1} = \frac{y}{2}$ 。求 $\frac{d u_{1}}{d y}$ 。

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解题步骤 1.1.1

对 $\frac{y}{2}$ 求导。

$\frac{d}{d y} [\frac{y}{2}]$

解题步骤 1.1.2

因为 $\frac{1}{2}$ 对于 $y$ 是常数，所以 $\frac{y}{2}$ 对 $y$ 的导数是 $\frac{1}{2} \frac{d}{d y} [y]$ 。

$\frac{1}{2} \frac{d}{d y} [y]$

解题步骤 1.1.3

使用幂法则求微分，根据该法则， $\frac{d}{d y} [y^{n}]$ 等于 $n y^{n - 1}$ ，其中 $n = 1$ 。

$\frac{1}{2} \cdot 1$

解题步骤 1.1.4

将 $\frac{1}{2}$ 乘以 $1$ 。

$\frac{1}{2}$

解题步骤 1.2

使用 $u_{1}$ 和 $d u_{1}$ 重写该问题。

$\int \cos^{6} (u_{1}) \frac{1}{\frac{1}{2}} d u_{1}$

解题步骤 2

化简。

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解题步骤 2.1

乘以分数的倒数从而实现除以 $\frac{1}{2}$ 。

$\int \cos^{6} (u_{1}) (1 \cdot 2) d u_{1}$

解题步骤 2.2

将 $2$ 乘以 $1$ 。

$\int \cos^{6} (u_{1}) \cdot 2 d u_{1}$

解题步骤 2.3

将 $2$ 移到 $\cos^{6} (u_{1})$ 的左侧。

$\int 2 \cos^{6} (u_{1}) d u_{1}$

解题步骤 3

由于 $2$ 对于 $u_{1}$ 是常数，所以将 $2$ 移到积分外。

$2 \int \cos^{6} (u_{1}) d u_{1}$

解题步骤 4

通过提取公因式进行化简。

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解题步骤 4.1

从 $6$ 中分解出因数 $2$ 。

$2 \int {\cos (u_{1})}^{2 (3)} d u_{1}$

解题步骤 4.2

将 ${\cos (u_{1})}^{2 (3)}$ 重写为乘方形式。

$2 \int {(\cos^{2} (u_{1}))}^{3} d u_{1}$

解题步骤 5

使用半角公式将 $\frac{1 + \cos (2 u_{1})}{2}$ 重新书写为 $\cos^{2} (u_{1})$ 的形式。

$2 \int {(\frac{1 + \cos (2 u_{1})}{2})}^{3} d u_{1}$

解题步骤 6

使 $u_{2} = 2 u_{1}$ 。然后使 $d u_{2} = 2 d u_{1}$ ，以便 $\frac{1}{2} d u_{2} = d u_{1}$ 。使用 $u_{2}$ 和 $d$ $u_{2}$ 进行重写。

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解题步骤 6.1

设 $u_{2} = 2 u_{1}$ 。求 $\frac{d u_{2}}{d u_{1}}$ 。

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解题步骤 6.1.1

对 $2 u_{1}$ 求导。

$\frac{d}{d u_{1}} [2 u_{1}]$

解题步骤 6.1.2

因为 $2$ 对于 $u_{1}$ 是常数，所以 $2 u_{1}$ 对 $u_{1}$ 的导数是 $2 \frac{d}{d u_{1}} [u_{1}]$ 。

$2 \frac{d}{d u_{1}} [u_{1}]$

解题步骤 6.1.3

使用幂法则求微分，根据该法则， $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ 等于 $n {u_{1}}^{n - 1}$ ，其中 $n = 1$ 。

$2 \cdot 1$

解题步骤 6.1.4

将 $2$ 乘以 $1$ 。

$2$

解题步骤 6.2

使用 $u_{2}$ 和 $d u_{2}$ 重写该问题。

$2 \int {(\frac{1 + \cos (u_{2})}{2})}^{3} \frac{1}{2} d u_{2}$

解题步骤 7

由于 $\frac{1}{2}$ 对于 $u_{2}$ 是常数，所以将 $\frac{1}{2}$ 移到积分外。

$2 (\frac{1}{2} \int {(\frac{1 + \cos (u_{2})}{2})}^{3} d u_{2})$

解题步骤 8

通过相乘进行化简。

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解题步骤 8.1

化简。

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解题步骤 8.1.1

组合 $\frac{1}{2}$ 和 $2$ 。

$\frac{2}{2} \int {(\frac{1 + \cos (u_{2})}{2})}^{3} d u_{2}$

解题步骤 8.1.2

约去 $2$ 的公因数。

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解题步骤 8.1.2.1

约去公因数。

$\frac{2}{2} \int {(\frac{1 + \cos (u_{2})}{2})}^{3} d u_{2}$

解题步骤 8.1.2.2

重写表达式。

$1 \int {(\frac{1 + \cos (u_{2})}{2})}^{3} d u_{2}$

解题步骤 8.1.3

将 $\int {(\frac{1 + \cos (u_{2})}{2})}^{3} d u_{2}$ 乘以 $1$ 。

$\int {(\frac{1 + \cos (u_{2})}{2})}^{3} d u_{2}$

解题步骤 8.2

将 $\frac{1 + \cos (u_{2})}{2}$ 重写为乘积形式。

$\int {(\frac{1}{2} \cdot (1 + \cos (u_{2})))}^{3} d u_{2}$

解题步骤 8.3

展开 ${(\frac{1}{2} \cdot (1 + \cos (u_{2})))}^{3}$ 。

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解题步骤 8.3.1

将幂重写为乘积形式。

$\int \frac{1}{2} \cdot (1 + \cos (u_{2})) {(\frac{1}{2} \cdot (1 + \cos (u_{2})))}^{2} d u_{2}$

解题步骤 8.3.2

将幂重写为乘积形式。

$\int \frac{1}{2} \cdot (1 + \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2})))) d u_{2}$

解题步骤 8.3.3

运用分配律。

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2})))) d u_{2}$

解题步骤 8.3.4

运用分配律。

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) ((\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot (1 + \cos (u_{2})))) d u_{2}$

解题步骤 8.3.5

运用分配律。

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) ((\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.6

运用分配律。

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.7

运用分配律。

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2})) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.8

运用分配律。

解题步骤 8.3.9

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.10

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.11

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.12

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.13

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.14

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.15

运用分配律。

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.16

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.17

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.18

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.19

移动 $\frac{1}{2}$ 。

$\int 1 \cdot \frac{1}{2} (1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.20

移动括号。

$\int 1 \cdot \frac{1}{2} (1 \cdot 1 \frac{1}{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.21

移动括号。

$\int 1 \cdot \frac{1}{2} (1 \cdot 1) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.22

移动 $\frac{1}{2}$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.23

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.24

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.25

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2} \frac{1}{2}) \cdot \cos (u_{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.26

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.27

移动 $\frac{1}{2}$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.28

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.29

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.30

移动 $\cos (u_{2})$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1 \cos (u_{2}) \cdot \frac{1}{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.31

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.32

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2} \cos (u_{2})) \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.33

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.34

移动 $\frac{1}{2}$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.35

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.36

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2}) \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.37

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2})) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.38

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.39

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.40

移动 $\frac{1}{2}$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.41

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot 1 \frac{1}{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.42

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot 1) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.43

移动 $\cos (u_{2})$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot 1 \cdot 1 \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.44

移动 $\frac{1}{2}$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.45

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{2}))) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.46

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2} \frac{1}{2}) \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.47

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2}) \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.48

移动 $\cos (u_{2})$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.49

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1)) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.50

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2})) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.51

移动 $\cos (u_{2})$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot 1 \cos (u_{2}) \cdot \frac{1}{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.52

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2}) + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.53

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2} \cos (u_{2})) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.54

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot \cos (u_{2}) (1 \cdot \frac{1}{2}) \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.55

移动 $\cos (u_{2})$ 。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + \frac{1}{2} \cdot 1 \cos (u_{2}) \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.56

将 $\frac{1}{2}$ 和 $1$ 重新排序。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}))) d u_{2}$

解题步骤 8.3.57

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2}) \cdot \cos (u_{2}) d u_{2}$

解题步骤 8.3.58

移动括号。

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{2}) + 1 \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{2}) (\frac{1}{2} \cdot \cos (u_{2})) \frac{1}{2} \cdot \cos (u_{2}) d u_{2}$

解题步骤 8.3.59

将 $1$ 乘以 $1$ 。

$\int 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.60

将 $1$ 乘以 $1$ 。

$\int 1 (\frac{1}{2}) \frac{1}{2} \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.61

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.62

将 $\frac{1}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{2 \cdot 2} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.63

将 $2$ 乘以 $2$ 。

$\int \frac{1}{4} \cdot \frac{1}{2} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.64

将 $\frac{1}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{4 \cdot 2} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.65

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.66

将 $1$ 乘以 $1$ 。

$\int \frac{1}{8} + 1 (\frac{1}{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.67

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.68

将 $\frac{1}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{1}{2 \cdot 2} \cdot \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.69

将 $2$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{1}{4} \cdot \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.70

将 $\frac{1}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{1}{4 \cdot 2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.71

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{1}{8} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.72

组合 $\frac{1}{8}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.73

将 $1$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.74

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.75

将 $\frac{1}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{2 \cdot 2} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.76

将 $2$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{4} \cos (u_{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.77

组合 $\frac{1}{4}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{4} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.78

将 $\frac{\cos (u_{2})}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{4 \cdot 2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.79

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.80

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.81

将 $\frac{1}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{2 \cdot 2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.82

将 $2$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{4} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.83

组合 $\frac{1}{4}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{4} \cdot \frac{1}{2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.84

将 $\frac{\cos (u_{2})}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{4 \cdot 2} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.85

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} \cos (u_{2}) + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.86

组合 $\frac{\cos (u_{2})}{8}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2}) \cos (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.87

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.88

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos^{1} (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.89

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{{\cos (u_{2})}^{1 + 1}}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.90

将 $1$ 和 $1$ 相加。

$\int \frac{1}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.91

将 $\frac{\cos (u_{2})}{8}$ 和 $\frac{\cos (u_{2})}{8}$ 相加。

$\int \frac{1}{8} + 2 \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.92

组合 $2$ 和 $\frac{\cos (u_{2})}{8}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.93

将 $1$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.94

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.95

组合 $\frac{1}{2}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{2} \cdot \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.96

将 $\frac{\cos (u_{2})}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{2 \cdot 2} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.97

将 $2$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{4} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.98

将 $\frac{\cos (u_{2})}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{4 \cdot 2} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.99

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.100

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.101

组合 $\frac{1}{2}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{2} \cdot \frac{1}{2} \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.102

将 $\frac{\cos (u_{2})}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{2 \cdot 2} \cdot \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.103

将 $2$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{4} \cdot \frac{1}{2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.104

将 $\frac{\cos (u_{2})}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{4 \cdot 2} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.105

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} \cos (u_{2}) + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.106

组合 $\frac{\cos (u_{2})}{8}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos (u_{2}) \cos (u_{2})}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.107

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos (u_{2})}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.108

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos^{1} (u_{2})}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.109

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{{\cos (u_{2})}^{1 + 1}}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.110

将 $1$ 和 $1$ 相加。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.111

将 $\frac{1}{2}$ 乘以 $1$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.112

组合 $\frac{1}{2}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{2} \cdot \frac{1}{2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.113

将 $\frac{\cos (u_{2})}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{2 \cdot 2} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.114

将 $2$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{4} \cos (u_{2}) \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.115

组合 $\frac{\cos (u_{2})}{4}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2}) \cos (u_{2})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.116

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos (u_{2})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.117

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos^{1} (u_{2})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.118

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{{\cos (u_{2})}^{1 + 1}}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.119

将 $1$ 和 $1$ 相加。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.120

将 $\frac{\cos^{2} (u_{2})}{4}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{4 \cdot 2} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.121

将 $4$ 乘以 $2$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.122

组合 $\frac{1}{2}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{2} \cdot \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.123

将 $\frac{\cos (u_{2})}{2}$ 乘以 $\frac{1}{2}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{2 \cdot 2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.124

将 $2$ 乘以 $2$ 。

解题步骤 8.3.125

组合 $\frac{\cos (u_{2})}{4}$ 和 $\cos (u_{2})$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2}) \cos (u_{2})}{4} \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.126

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos (u_{2})}{4} \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.127

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{1} (u_{2}) \cos^{1} (u_{2})}{4} \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.128

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{{\cos (u_{2})}^{1 + 1}}{4} \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.129

将 $1$ 和 $1$ 相加。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{4} \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

解题步骤 8.3.130

将 $\frac{\cos^{2} (u_{2})}{4}$ 乘以 $\frac{1}{2}$ 。

解题步骤 8.3.131

将 $4$ 乘以 $2$ 。

解题步骤 8.3.132

组合 $\frac{\cos^{2} (u_{2})}{8}$ 和 $\cos (u_{2})$ 。

解题步骤 8.3.133

对 $\cos (u_{2})$ 进行 $1$ 次方运算。

解题步骤 8.3.134

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

解题步骤 8.3.135

将 $2$ 和 $1$ 相加。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} d u_{2}$

解题步骤 8.3.136

将 $\frac{\cos^{2} (u_{2})}{8}$ 和 $\frac{\cos^{2} (u_{2})}{8}$ 相加。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + 2 \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} d u_{2}$

解题步骤 8.3.137

组合 $2$ 和 $\frac{\cos^{2} (u_{2})}{8}$ 。

$\int \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} d u_{2}$

解题步骤 8.3.138

将 $\frac{2 \cos (u_{2})}{8}$ 和 $\frac{\cos^{2} (u_{2})}{8}$ 重新排序。

$\int \frac{1}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} d u_{2}$

解题步骤 8.3.139

将 $\frac{1}{8}$ 和 $\frac{\cos^{2} (u_{2})}{8}$ 重新排序。

$\int \frac{\cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} d u_{2}$

解题步骤 8.3.140

将 $\frac{2 \cos^{2} (u_{2})}{8}$ 和 $\frac{\cos^{3} (u_{2})}{8}$ 重新排序。

$\int \frac{\cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} d u_{2}$

解题步骤 8.3.141

移动 $\frac{\cos (u_{2})}{8}$ 。

$\int \frac{\cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{\cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.142

移动 $\frac{2 \cos (u_{2})}{8}$ 。

$\int \frac{\cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{\cos^{3} (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.143

移动 $\frac{1}{8}$ 。

$\int \frac{\cos^{2} (u_{2})}{8} + \frac{\cos^{3} (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.144

将 $\frac{\cos^{2} (u_{2})}{8}$ 和 $\frac{\cos^{3} (u_{2})}{8}$ 重新排序。

$\int \frac{\cos^{3} (u_{2})}{8} + \frac{\cos^{2} (u_{2})}{8} + \frac{2 \cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.145

在公分母上合并分子。

$\int \frac{\cos^{3} (u_{2})}{8} + \frac{\cos^{2} (u_{2}) + 2 \cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.146

将 $\cos^{2} (u_{2})$ 和 $2 \cos^{2} (u_{2})$ 相加。

$\int \frac{\cos^{3} (u_{2})}{8} + \frac{3 \cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2})}{8} + \frac{\cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.147

在公分母上合并分子。

$\int \frac{\cos^{3} (u_{2})}{8} + \frac{3 \cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{2 \cos (u_{2}) + \cos (u_{2})}{8} d u_{2}$

解题步骤 8.3.148

将 $2 \cos (u_{2})$ 和 $\cos (u_{2})$ 相加。

$\int \frac{\cos^{3} (u_{2})}{8} + \frac{3 \cos^{2} (u_{2})}{8} + \frac{1}{8} + \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 9

将单个积分拆分为多个积分。

$\int \frac{\cos^{3} (u_{2})}{8} d u_{2} + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 10

由于 $\frac{1}{8}$ 对于 $u_{2}$ 是常数，所以将 $\frac{1}{8}$ 移到积分外。

$\frac{1}{8} \int \cos^{3} (u_{2}) d u_{2} + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 11

因式分解出 $\cos^{2} (u_{2})$ 。

$\frac{1}{8} \int \cos^{2} (u_{2}) \cos (u_{2}) d u_{2} + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 12

使用勾股定理，将 $\cos^{2} (u_{2})$ 重写成 $1 - \sin^{2} (u_{2})$ 的形式。

$\frac{1}{8} \int (1 - \sin^{2} (u_{2})) \cos (u_{2}) d u_{2} + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 13

使 $u_{3} = \sin (u_{2})$ 。然后使 $d u_{3} = \cos (u_{2}) d u_{2}$ ，以便 $\frac{1}{\cos (u_{2})} d u_{3} = d u_{2}$ 。使用 $u_{3}$ 和 $d$ $u_{3}$ 进行重写。

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解题步骤 13.1

设 $u_{3} = \sin (u_{2})$ 。求 $\frac{d u_{3}}{d u_{2}}$ 。

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解题步骤 13.1.1

对 $\sin (u_{2})$ 求导。

$\frac{d}{d u_{2}} [\sin (u_{2})]$

解题步骤 13.1.2

$\sin (u_{2})$ 对 $u_{2}$ 的导数为 $\cos (u_{2})$ 。

$\cos (u_{2})$

解题步骤 13.2

使用 $u_{3}$ 和 $d u_{3}$ 重写该问题。

$\frac{1}{8} \int 1 - {u_{3}}^{2} d u_{3} + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 14

将单个积分拆分为多个积分。

$\frac{1}{8} (\int d u_{3} + \int - {u_{3}}^{2} d u_{3}) + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 15

应用常数不变法则。

$\frac{1}{8} (u_{3} + C + \int - {u_{3}}^{2} d u_{3}) + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 16

由于 $- 1$ 对于 $u_{3}$ 是常数，所以将 $- 1$ 移到积分外。

$\frac{1}{8} (u_{3} + C - \int {u_{3}}^{2} d u_{3}) + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 17

根据幂法则， ${u_{3}}^{2}$ 对 $u_{3}$ 的积分是 $\frac{1}{3} {u_{3}}^{3}$ 。

$\frac{1}{8} (u_{3} + C - (\frac{1}{3} {u_{3}}^{3} + C)) + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 18

组合 $\frac{1}{3}$ 和 ${u_{3}}^{3}$ 。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \int \frac{3 \cos^{2} (u_{2})}{8} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 19

由于 $\frac{3}{8}$ 对于 $u_{2}$ 是常数，所以将 $\frac{3}{8}$ 移到积分外。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{8} \int \cos^{2} (u_{2}) d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 20

使用半角公式将 $\frac{1 + \cos (2 u_{2})}{2}$ 重新书写为 $\cos^{2} (u_{2})$ 的形式。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{8} \int \frac{1 + \cos (2 u_{2})}{2} d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 21

由于 $\frac{1}{2}$ 对于 $u_{2}$ 是常数，所以将 $\frac{1}{2}$ 移到积分外。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{8} (\frac{1}{2} \int 1 + \cos (2 u_{2}) d u_{2}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 22

化简。

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解题步骤 22.1

将 $\frac{1}{2}$ 乘以 $\frac{3}{8}$ 。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{2 \cdot 8} \int 1 + \cos (2 u_{2}) d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 22.2

将 $2$ 乘以 $8$ 。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} \int 1 + \cos (2 u_{2}) d u_{2} + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 23

将单个积分拆分为多个积分。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (\int d u_{2} + \int \cos (2 u_{2}) d u_{2}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 24

应用常数不变法则。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \int \cos (2 u_{2}) d u_{2}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 25

使 $u_{4} = 2 u_{2}$ 。然后使 $d u_{4} = 2 d u_{2}$ ，以便 $\frac{1}{2} d u_{4} = d u_{2}$ 。使用 $u_{4}$ 和 $d$ $u_{4}$ 进行重写。

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解题步骤 25.1

设 $u_{4} = 2 u_{2}$ 。求 $\frac{d u_{4}}{d u_{2}}$ 。

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解题步骤 25.1.1

对 $2 u_{2}$ 求导。

$\frac{d}{d u_{2}} [2 u_{2}]$

解题步骤 25.1.2

因为 $2$ 对于 $u_{2}$ 是常数，所以 $2 u_{2}$ 对 $u_{2}$ 的导数是 $2 \frac{d}{d u_{2}} [u_{2}]$ 。

$2 \frac{d}{d u_{2}} [u_{2}]$

解题步骤 25.1.3

使用幂法则求微分，根据该法则， $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ 等于 $n {u_{2}}^{n - 1}$ ，其中 $n = 1$ 。

$2 \cdot 1$

解题步骤 25.1.4

将 $2$ 乘以 $1$ 。

$2$

解题步骤 25.2

使用 $u_{4}$ 和 $d u_{4}$ 重写该问题。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \int \cos (u_{4}) \frac{1}{2} d u_{4}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 26

组合 $\cos (u_{4})$ 和 $\frac{1}{2}$ 。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \int \frac{\cos (u_{4})}{2} d u_{4}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 27

由于 $\frac{1}{2}$ 对于 $u_{4}$ 是常数，所以将 $\frac{1}{2}$ 移到积分外。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} \int \cos (u_{4}) d u_{4}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 28

$\cos (u_{4})$ 对 $u_{4}$ 的积分为 $\sin (u_{4})$ 。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 29

应用常数不变法则。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \frac{1}{8} u_{2} + C + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

解题步骤 30

由于 $\frac{3}{8}$ 对于 $u_{2}$ 是常数，所以将 $\frac{3}{8}$ 移到积分外。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \frac{1}{8} u_{2} + C + \frac{3}{8} \int \cos (u_{2}) d u_{2}$

解题步骤 31

$\cos (u_{2})$ 对 $u_{2}$ 的积分为 $\sin (u_{2})$ 。

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \frac{1}{8} u_{2} + C + \frac{3}{8} (\sin (u_{2}) + C)$

解题步骤 32

化简。

$\frac{u_{3}}{8} - \frac{{u_{3}}^{3}}{24} + \frac{3 u_{2}}{16} + \frac{3 \sin (u_{4})}{32} + \frac{u_{2}}{8} + \frac{3}{8} \sin (u_{2}) + C$

解题步骤 33

重新排序项。

$\frac{1}{8} u_{3} - \frac{1}{24} {u_{3}}^{3} + \frac{3}{16} u_{2} + \frac{3}{32} \sin (u_{4}) + \frac{1}{8} u_{2} + \frac{3}{8} \sin (u_{2}) + C$

解题步骤 34

代回替换每一个积分法替换变量。

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解题步骤 34.1

使用 $\sin (u_{2})$ 替换所有出现的 $u_{3}$ 。

$\frac{1}{8} \sin (u_{2}) - \frac{1}{24} \sin^{3} (u_{2}) + \frac{3}{16} u_{2} + \frac{3}{32} \sin (u_{4}) + \frac{1}{8} u_{2} + \frac{3}{8} \sin (u_{2}) + C$

解题步骤 34.2

使用 $2 u_{1}$ 替换所有出现的 $u_{2}$ 。

$\frac{1}{8} \sin (2 u_{1}) - \frac{1}{24} \sin^{3} (2 u_{1}) + \frac{3}{16} (2 u_{1}) + \frac{3}{32} \sin (u_{4}) + \frac{1}{8} (2 u_{1}) + \frac{3}{8} \sin (2 u_{1}) + C$

解题步骤 34.3

使用 $2 u_{2}$ 替换所有出现的 $u_{4}$ 。

$\frac{1}{8} \sin (2 u_{1}) - \frac{1}{24} \sin^{3} (2 u_{1}) + \frac{3}{16} (2 u_{1}) + \frac{3}{32} \sin (2 u_{2}) + \frac{1}{8} (2 u_{1}) + \frac{3}{8} \sin (2 u_{1}) + C$

解题步骤 34.4

使用 $\frac{y}{2}$ 替换所有出现的 $u_{1}$ 。

$\frac{1}{8} \sin (2 \frac{y}{2}) - \frac{1}{24} \sin^{3} (2 \frac{y}{2}) + \frac{3}{16} (2 \frac{y}{2}) + \frac{3}{32} \sin (2 u_{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 34.5

使用 $2 u_{1}$ 替换所有出现的 $u_{2}$ 。

$\frac{1}{8} \sin (2 \frac{y}{2}) - \frac{1}{24} \sin^{3} (2 \frac{y}{2}) + \frac{3}{16} (2 \frac{y}{2}) + \frac{3}{32} \sin (2 (2 u_{1})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 34.6

使用 $\frac{y}{2}$ 替换所有出现的 $u_{1}$ 。

$\frac{1}{8} \sin (2 \frac{y}{2}) - \frac{1}{24} \sin^{3} (2 \frac{y}{2}) + \frac{3}{16} (2 \frac{y}{2}) + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35

化简。

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解题步骤 35.1

组合 $2$ 和 $\frac{y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{1}{24} \sin^{3} (2 \frac{y}{2}) + \frac{3}{16} (2 \frac{y}{2}) + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.2

组合 $2$ 和 $\frac{y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{1}{24} \sin^{3} (\frac{2 y}{2}) + \frac{3}{16} (2 \frac{y}{2}) + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.3

组合 $\sin^{3} (\frac{2 y}{2})$ 和 $\frac{1}{24}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{3}{16} (2 \frac{y}{2}) + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.4

组合 $2$ 和 $\frac{y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{3}{16} \cdot \frac{2 y}{2} + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.5

将 $\frac{3}{16}$ 乘以 $\frac{2 y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{3 (2 y)}{16 \cdot 2} + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.6

将 $2$ 乘以 $3$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{16 \cdot 2} + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.7

将 $16$ 乘以 $2$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.8

组合 $2$ 和 $\frac{y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (2 \frac{2 y}{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.9

组合 $2$ 和 $\frac{2 y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{2 (2 y)}{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.10

将 $2$ 乘以 $2$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.11

组合 $2$ 和 $\frac{y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{1}{8} \cdot \frac{2 y}{2} + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.12

将 $\frac{1}{8}$ 乘以 $\frac{2 y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{8 \cdot 2} + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.13

将 $8$ 乘以 $2$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

解题步骤 35.14

组合 $2$ 和 $\frac{y}{2}$ 。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36

化简。

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解题步骤 36.1

约去 $2$ 的公因数。

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解题步骤 36.1.1

约去公因数。

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.1.2

用 $y$ 除以 $1$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.2

约去 $2$ 的公因数。

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解题步骤 36.2.1

约去公因数。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.2.2

用 $y$ 除以 $1$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.3

约去 $6$ 和 $32$ 的公因数。

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解题步骤 36.3.1

从 $6 y$ 中分解出因数 $2$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{2 (3 y)}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.3.2

约去公因数。

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解题步骤 36.3.2.1

从 $32$ 中分解出因数 $2$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{2 (3 y)}{2 (16)} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.3.2.2

约去公因数。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{2 (3 y)}{2 \cdot 16} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.3.2.3

重写表达式。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.4

约去 $4$ 和 $2$ 的公因数。

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解题步骤 36.4.1

从 $4 y$ 中分解出因数 $2$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (\frac{2 (2 y)}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.4.2

约去公因数。

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解题步骤 36.4.2.1

从 $2$ 中分解出因数 $2$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (\frac{2 (2 y)}{2 (1)}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.4.2.2

约去公因数。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (\frac{2 (2 y)}{2 \cdot 1}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.4.2.3

重写表达式。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (\frac{2 y}{1}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.4.2.4

用 $2 y$ 除以 $1$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.5

约去 $2$ 和 $16$ 的公因数。

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解题步骤 36.5.1

从 $2 y$ 中分解出因数 $2$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{2 (y)}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.5.2

约去公因数。

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解题步骤 36.5.2.1

从 $16$ 中分解出因数 $2$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{2 y}{2 \cdot 8} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.5.2.2

约去公因数。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{2 y}{2 \cdot 8} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.5.2.3

重写表达式。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{y}{8} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.6

约去 $2$ 的公因数。

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解题步骤 36.6.1

约去公因数。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{y}{8} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

解题步骤 36.6.2

用 $y$ 除以 $1$ 。

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{3 y}{16} + \frac{3}{32} \sin (2 y) + \frac{y}{8} + \frac{3}{8} \sin (y) + C$

解题步骤 36.7

重新排序项。

$\frac{1}{8} \sin (y) - \frac{1}{24} \sin^{3} (y) + \frac{3}{16} y + \frac{3}{32} \sin (2 y) + \frac{1}{8} y + \frac{3}{8} \sin (y) + C$

微积分学 示例

微积分学示例