微积分学示例

\frac{d y}{d x} + tan (x) y = 1

$\frac{d y}{d x} + \tan (x) y = 1$

解题步骤 1

积分因数由公式

e^{\int P (x) d x}

$e^{\int P (x) d x}$ 定义，其中

P (x) = tan (x)

$P (x) = \tan (x)$ 。

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解题步骤 1.1

建立积分。

e^{\int tan (x) d x}

$e^{\int \tan (x) d x}$

解题步骤 1.2

tan (x)

$\tan (x)$ 对

x

$x$ 的积分为

ln (| sec (x) |)

$\ln (| \sec (x) |)$ 。

e^{ln (| sec (x) |) + C}

$e^{\ln (| \sec (x) |) + C}$

解题步骤 1.3

去掉积分常数。

e^{ln (sec (x))}

$e^{\ln (\sec (x))}$

解题步骤 1.4

指数函数和对数函数互为反函数。

sec (x)

$\sec (x)$

sec (x)

$\sec (x)$

解题步骤 2

每一项乘以积分因数

sec (x)

$\sec (x)$ 。

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解题步骤 2.1

每一项乘以

sec (x)

$\sec (x)$ 。

sec (x) \frac{d y}{d x} + sec (x) (tan (x) y) = sec (x) \cdot 1

$\sec (x) \frac{d y}{d x} + \sec (x) (\tan (x) y) = \sec (x) \cdot 1$

解题步骤 2.2

将

sec (x)

$\sec (x)$ 乘以

1

$1$ 。

sec (x) \frac{d y}{d x} + sec (x) tan (x) y = sec (x)

$\sec (x) \frac{d y}{d x} + \sec (x) \tan (x) y = \sec (x)$

解题步骤 2.3

将

sec (x) \frac{d y}{d x} + sec (x) tan (x) y = sec (x)

$\sec (x) \frac{d y}{d x} + \sec (x) \tan (x) y = \sec (x)$ 中的因式重新排序。

sec (x) \frac{d y}{d x} + y sec (x) tan (x) = sec (x)

$\sec (x) \frac{d y}{d x} + y \sec (x) \tan (x) = \sec (x)$

sec (x) \frac{d y}{d x} + y sec (x) tan (x) = sec (x)

$\sec (x) \frac{d y}{d x} + y \sec (x) \tan (x) = \sec (x)$

解题步骤 3

将左边重写为对积求导的结果。

\frac{d}{d x} [sec (x) y] = sec (x)

$\frac{d}{d x} [\sec (x) y] = \sec (x)$

解题步骤 4

在两边建立积分。

\int \frac{d}{d x} [sec (x) y] d x = \int sec (x) d x

$\int \frac{d}{d x} [\sec (x) y] d x = \int \sec (x) d x$

解题步骤 5

对左边积分。

sec (x) y = \int sec (x) d x

$\sec (x) y = \int \sec (x) d x$

解题步骤 6

sec (x)

$\sec (x)$ 对

x

$x$ 的积分为

ln (| sec (x) + tan (x) |)

$\ln (| \sec (x) + \tan (x) |)$ 。

sec (x) y = ln (| sec (x) + tan (x) |) + C

$\sec (x) y = \ln (| \sec (x) + \tan (x) |) + C$

解题步骤 7

将

sec (x) y = ln (| sec (x) + tan (x) |) + C

$\sec (x) y = \ln (| \sec (x) + \tan (x) |) + C$ 中的每一项除以

sec (x)

$\sec (x)$ 并化简。

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解题步骤 7.1

将

sec (x) y = ln (| sec (x) + tan (x) |) + C

$\sec (x) y = \ln (| \sec (x) + \tan (x) |) + C$ 中的每一项都除以

sec (x)

$\sec (x)$ 。

\frac{sec (x) y}{sec (x)} = \frac{ln (| sec (x) + tan (x) |)}{sec (x)} + \frac{C}{sec (x)}

$\frac{\sec (x) y}{\sec (x)} = \frac{\ln (| \sec (x) + \tan (x) |)}{\sec (x)} + \frac{C}{\sec (x)}$

解题步骤 7.2

化简左边。

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解题步骤 7.2.1

约去

sec (x)

$\sec (x)$ 的公因数。

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解题步骤 7.2.1.1

约去公因数。

\frac{sec (x) y}{sec (x)} = \frac{ln (| sec (x) + tan (x) |)}{sec (x)} + \frac{C}{sec (x)}

$\frac{\sec (x) y}{\sec (x)} = \frac{\ln (| \sec (x) + \tan (x) |)}{\sec (x)} + \frac{C}{\sec (x)}$

解题步骤 7.2.1.2

用

y

$y$ 除以

1

$1$ 。

y = \frac{ln (| sec (x) + tan (x) |)}{sec (x)} + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{\sec (x)} + \frac{C}{\sec (x)}$

y = \frac{ln (| sec (x) + tan (x) |)}{sec (x)} + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{\sec (x)} + \frac{C}{\sec (x)}$

y = \frac{ln (| sec (x) + tan (x) |)}{sec (x)} + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{\sec (x)} + \frac{C}{\sec (x)}$

解题步骤 7.3

化简右边。

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解题步骤 7.3.1

化简每一项。

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解题步骤 7.3.1.1

分离分数。

y = \frac{ln (| sec (x) + tan (x) |)}{1} \cdot \frac{1}{sec (x)} + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{1} \cdot \frac{1}{\sec (x)} + \frac{C}{\sec (x)}$

解题步骤 7.3.1.2

将

sec (x)

$\sec (x)$ 重写为正弦和余弦形式。

y = \frac{ln (| sec (x) + tan (x) |)}{1} \cdot \frac{1}{\frac{1}{cos (x)}} + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{1} \cdot \frac{1}{\frac{1}{\cos (x)}} + \frac{C}{\sec (x)}$

解题步骤 7.3.1.3

乘以分数的倒数从而实现除以

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ 。

y = \frac{ln (| sec (x) + tan (x) |)}{1} (1 cos (x)) + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{1} (1 \cos (x)) + \frac{C}{\sec (x)}$

解题步骤 7.3.1.4

将

cos (x)

$\cos (x)$ 乘以

1

$1$ 。

y = \frac{ln (| sec (x) + tan (x) |)}{1} cos (x) + \frac{C}{sec (x)}

$y = \frac{\ln (| \sec (x) + \tan (x) |)}{1} \cos (x) + \frac{C}{\sec (x)}$

解题步骤 7.3.1.5

用

ln (| sec (x) + tan (x) |)

$\ln (| \sec (x) + \tan (x) |)$ 除以

1

$1$ 。

y = ln (| sec (x) + tan (x) |) cos (x) + \frac{C}{sec (x)}

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + \frac{C}{\sec (x)}$

解题步骤 7.3.1.6

分离分数。

y = ln (| sec (x) + tan (x) |) cos (x) + \frac{C}{1} \cdot \frac{1}{sec (x)}

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + \frac{C}{1} \cdot \frac{1}{\sec (x)}$

解题步骤 7.3.1.7

将

sec (x)

$\sec (x)$ 重写为正弦和余弦形式。

y = ln (| sec (x) + tan (x) |) cos (x) + \frac{C}{1} \cdot \frac{1}{\frac{1}{cos (x)}}

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + \frac{C}{1} \cdot \frac{1}{\frac{1}{\cos (x)}}$

解题步骤 7.3.1.8

乘以分数的倒数从而实现除以

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ 。

y = ln (| sec (x) + tan (x) |) cos (x) + \frac{C}{1} (1 cos (x))

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + \frac{C}{1} (1 \cos (x))$

解题步骤 7.3.1.9

将

cos (x)

$\cos (x)$ 乘以

1

$1$ 。

y = ln (| sec (x) + tan (x) |) cos (x) + \frac{C}{1} cos (x)

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + \frac{C}{1} \cos (x)$

解题步骤 7.3.1.10

用

C

$C$ 除以

1

$1$ 。

y = ln (| sec (x) + tan (x) |) cos (x) + C cos (x)

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + C \cos (x)$

y = ln (| sec (x) + tan (x) |) cos (x) + C cos (x)

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + C \cos (x)$

y = ln (| sec (x) + tan (x) |) cos (x) + C cos (x)

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + C \cos (x)$

y = ln (| sec (x) + tan (x) |) cos (x) + C cos (x)

$y = \ln (| \sec (x) + \tan (x) |) \cos (x) + C \cos (x)$

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微积分学 示例

微积分学示例