Calculus Examples

$\int_{0}^{2 π} \frac{1}{2} \cdot {(3 + \sin (4 x))}^{2} d x$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{0}^{2 π} {(3 + \sin (4 x))}^{2} d x$

Step 2

Let $u_{1} = 4 x$ . Then $d u_{1} = 4 d x$ , so $\frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 2.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \cdot 1$

Step 2.1.4

Multiply $4$ by $1$ .

$4$

Step 2.2

Substitute the lower limit in for $x$ in $u_{1} = 4 x$ .

$u_{lower} = 4 \cdot 0$

Step 2.3

Multiply $4$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u_{1} = 4 x$ .

$u_{upper} = 4 (2 π)$

Step 2.5

Multiply $2$ by $4$ .

$u_{upper} = 8 π$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 8 π$

Step 2.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\frac{1}{2} \int_{0}^{8 π} {(3 + \sin (u_{1}))}^{2} \frac{1}{4} d u_{1}$

Step 3

Combine ${(3 + \sin (u_{1}))}^{2}$ and $\frac{1}{4}$ .

$\frac{1}{2} \int_{0}^{8 π} \frac{{(3 + \sin (u_{1}))}^{2}}{4} d u_{1}$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} (\frac{1}{4} \int_{0}^{8 π} {(3 + \sin (u_{1}))}^{2} d u_{1})$

Step 5

Simplify by multiplying through.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$\frac{1}{4 \cdot 2} \int_{0}^{8 π} {(3 + \sin (u_{1}))}^{2} d u_{1}$

Step 5.1.2

Multiply $4$ by $2$ .

$\frac{1}{8} \int_{0}^{8 π} {(3 + \sin (u_{1}))}^{2} d u_{1}$

Step 5.2

Expand ${(3 + \sin (u_{1}))}^{2}$ .

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Step 5.2.1

Rewrite ${(3 + \sin (u_{1}))}^{2}$ as $(3 + \sin (u_{1})) (3 + \sin (u_{1}))$ .

$\frac{1}{8} \int_{0}^{8 π} (3 + \sin (u_{1})) (3 + \sin (u_{1})) d u_{1}$

Step 5.2.2

Apply the distributive property.

$\frac{1}{8} \int_{0}^{8 π} 3 (3 + \sin (u_{1})) + \sin (u_{1}) (3 + \sin (u_{1})) d u_{1}$

Step 5.2.3

Apply the distributive property.

$\frac{1}{8} \int_{0}^{8 π} 3 \cdot 3 + 3 \sin (u_{1}) + \sin (u_{1}) (3 + \sin (u_{1})) d u_{1}$

Step 5.2.4

Apply the distributive property.

$\frac{1}{8} \int_{0}^{8 π} 3 \cdot 3 + 3 \sin (u_{1}) + \sin (u_{1}) \cdot 3 + \sin (u_{1}) \sin (u_{1}) d u_{1}$

Step 5.2.5

Reorder $\sin (u_{1})$ and $3$ .

$\frac{1}{8} \int_{0}^{8 π} 3 \cdot 3 + 3 \sin (u_{1}) + 3 \cdot \sin (u_{1}) + \sin (u_{1}) \sin (u_{1}) d u_{1}$

Step 5.2.6

Multiply $3$ by $3$ .

$\frac{1}{8} \int_{0}^{8 π} 9 + 3 \sin (u_{1}) + 3 \sin (u_{1}) + \sin (u_{1}) \sin (u_{1}) d u_{1}$

Step 5.2.7

Raise $\sin (u_{1})$ to the power of $1$ .

$\frac{1}{8} \int_{0}^{8 π} 9 + 3 \sin (u_{1}) + 3 \sin (u_{1}) + \sin^{1} (u_{1}) \sin (u_{1}) d u_{1}$

Step 5.2.8

Raise $\sin (u_{1})$ to the power of $1$ .

$\frac{1}{8} \int_{0}^{8 π} 9 + 3 \sin (u_{1}) + 3 \sin (u_{1}) + \sin^{1} (u_{1}) \sin^{1} (u_{1}) d u_{1}$

Step 5.2.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{8} \int_{0}^{8 π} 9 + 3 \sin (u_{1}) + 3 \sin (u_{1}) + {\sin (u_{1})}^{1 + 1} d u_{1}$

Step 5.2.10

Add $1$ and $1$ .

$\frac{1}{8} \int_{0}^{8 π} 9 + 3 \sin (u_{1}) + 3 \sin (u_{1}) + \sin^{2} (u_{1}) d u_{1}$

Step 5.2.11

Add $3 \sin (u_{1})$ and $3 \sin (u_{1})$ .

$\frac{1}{8} \int_{0}^{8 π} 9 + 6 \sin (u_{1}) + \sin^{2} (u_{1}) d u_{1}$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{8} (\int_{0}^{8 π} 9 d u_{1} + \int_{0}^{8 π} 6 \sin (u_{1}) d u_{1} + \int_{0}^{8 π} \sin^{2} (u_{1}) d u_{1})$

Step 7

Apply the constant rule.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + \int_{0}^{8 π} 6 \sin (u_{1}) d u_{1} + \int_{0}^{8 π} \sin^{2} (u_{1}) d u_{1})$

Step 8

Since $6$ is constant with respect to $u_{1}$ , move $6$ out of the integral.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 \int_{0}^{8 π} \sin (u_{1}) d u_{1} + \int_{0}^{8 π} \sin^{2} (u_{1}) d u_{1})$

Step 9

The integral of $\sin (u_{1})$ with respect to $u_{1}$ is $- \cos (u_{1})$ .

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \int_{0}^{8 π} \sin^{2} (u_{1}) d u_{1})$

Step 10

Use the half-angle formula to rewrite $\sin^{2} (u_{1})$ as $\frac{1 - \cos (2 u_{1})}{2}$ .

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \int_{0}^{8 π} \frac{1 - \cos (2 u_{1})}{2} d u_{1})$

Step 11

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} \int_{0}^{8 π} 1 - \cos (2 u_{1}) d u_{1})$

Step 12

Split the single integral into multiple integrals.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (\int_{0}^{8 π} d u_{1} + \int_{0}^{8 π} - \cos (2 u_{1}) d u_{1}))$

Step 13

Apply the constant rule.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} + \int_{0}^{8 π} - \cos (2 u_{1}) d u_{1}))$

Step 14

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} - \int_{0}^{8 π} \cos (2 u_{1}) d u_{1}))$

Step 15

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 15.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 15.1.1

Differentiate $2 u_{1}$ .

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

Step 15.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

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

Step 15.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 15.1.4

Multiply $2$ by $1$ .

$2$

Step 15.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{lower} = 2 \cdot 0$

Step 15.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 15.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{upper} = 2 (8 π)$

Step 15.5

Multiply $8$ by $2$ .

$u_{upper} = 16 π$

Step 15.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 16 π$

Step 15.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} - \int_{0}^{16 π} \cos (u_{2}) \frac{1}{2} d u_{2}))$

Step 16

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} - \int_{0}^{16 π} \frac{\cos (u_{2})}{2} d u_{2}))$

Step 17

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} - (\frac{1}{2} \int_{0}^{16 π} \cos (u_{2}) d u_{2})))$

Step 18

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{8} (9 u_{1}]_{0}^{8 π} + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} - \frac{1}{2} \sin (u_{2})]_{0}^{16 π}))$

Step 19

Substitute and simplify.

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Step 19.1

Evaluate $9 u_{1}$ at $8 π$ and at $0$ .

$\frac{1}{8} ((9 (8 π)) - 9 \cdot 0 + 6 (- \cos (u_{1})]_{0}^{8 π}) + \frac{1}{2} (u_{1}]_{0}^{8 π} - \frac{1}{2} \sin (u_{2})]_{0}^{16 π}))$

Step 19.2

Evaluate $- \cos (u_{1})$ at $8 π$ and at $0$ .

$\frac{1}{8} (9 (8 π) - 9 \cdot 0 + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} (u_{1}]_{0}^{8 π} - \frac{1}{2} \sin (u_{2})]_{0}^{16 π}))$

Step 19.3

Evaluate $u_{1}$ at $8 π$ and at $0$ .

$\frac{1}{8} (9 (8 π) - 9 \cdot 0 + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} ((8 π) + 0 - \frac{1}{2} \sin (u_{2})]_{0}^{16 π}))$

Step 19.4

Evaluate $\sin (u_{2})$ at $16 π$ and at $0$ .

$\frac{1}{8} (9 (8 π) - 9 \cdot 0 + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} ((8 π) + 0 - \frac{1}{2} ((\sin (16 π)) - \sin (0))))$

Step 19.5

Simplify.

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Step 19.5.1

Multiply $8$ by $9$ .

$\frac{1}{8} (72 π - 9 \cdot 0 + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} ((8 π) + 0 - \frac{1}{2} ((\sin (16 π)) - \sin (0))))$

Step 19.5.2

Multiply $- 9$ by $0$ .

$\frac{1}{8} (72 π + 0 + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} ((8 π) + 0 - \frac{1}{2} ((\sin (16 π)) - \sin (0))))$

Step 19.5.3

Add $72 π$ and $0$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} ((8 π) + 0 - \frac{1}{2} ((\sin (16 π)) - \sin (0))))$

Step 19.5.4

Add $8 π$ and $0$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + \cos (0)) + \frac{1}{2} (8 π - \frac{1}{2} (\sin (16 π) - \sin (0))))$

Step 20

Simplify.

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Step 20.1

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + 1) + \frac{1}{2} (8 π - \frac{1}{2} (\sin (16 π) - \sin (0))))$

Step 20.2

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + 1) + \frac{1}{2} (8 π - \frac{1}{2} (\sin (16 π) - 0)))$

Step 20.3

Multiply $- 1$ by $0$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + 1) + \frac{1}{2} (8 π - \frac{1}{2} (\sin (16 π) + 0)))$

Step 20.4

Add $\sin (16 π)$ and $0$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + 1) + \frac{1}{2} (8 π - \frac{1}{2} \sin (16 π)))$

Step 20.5

Combine $\sin (16 π)$ and $\frac{1}{2}$ .

$\frac{1}{8} (72 π + 6 (- \cos (8 π) + 1) + \frac{1}{2} (8 π - \frac{\sin (16 π)}{2}))$

Step 21

Simplify.

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Step 21.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{1}{8} (72 π + 6 (- \cos (0) + 1) + \frac{1}{2} (8 π - \frac{\sin (16 π)}{2}))$

Step 21.2

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{8} (72 π + 6 (- 1 \cdot 1 + 1) + \frac{1}{2} (8 π - \frac{\sin (16 π)}{2}))$

Step 21.3

Multiply $- 1$ by $1$ .

$\frac{1}{8} (72 π + 6 (- 1 + 1) + \frac{1}{2} (8 π - \frac{\sin (16 π)}{2}))$

Step 21.4

Add $- 1$ and $1$ .

$\frac{1}{8} (72 π + 6 \cdot 0 + \frac{1}{2} (8 π - \frac{\sin (16 π)}{2}))$

Step 21.5

Multiply $6$ by $0$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (8 π - \frac{\sin (16 π)}{2}))$

Step 21.6

Simplify the numerator.

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Step 21.6.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (8 π - \frac{\sin (0)}{2}))$

Step 21.6.2

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (8 π - \frac{0}{2}))$

Step 21.7

Divide $0$ by $2$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (8 π - 0))$

Step 21.8

Multiply $- 1$ by $0$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (8 π + 0))$

Step 21.9

Add $8 π$ and $0$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (8 π))$

Step 21.10

Cancel the common factor of $2$ .

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Step 21.10.1

Factor $2$ out of $8 π$ .

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (2 (4 π)))$

Step 21.10.2

Cancel the common factor.

$\frac{1}{8} (72 π + 0 + \frac{1}{2} (2 (4 π)))$

Step 21.10.3

Rewrite the expression.

$\frac{1}{8} (72 π + 0 + 4 π)$

Step 21.11

Add $72 π$ and $0$ .

$\frac{1}{8} (72 π + 4 π)$

Step 21.12

Add $72 π$ and $4 π$ .

$\frac{1}{8} (76 π)$

Step 21.13

Cancel the common factor of $4$ .

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Step 21.13.1

Factor $4$ out of $8$ .

$\frac{1}{4 (2)} (76 π)$

Step 21.13.2

Factor $4$ out of $76 π$ .

$\frac{1}{4 (2)} (4 (19 π))$

Step 21.13.3

Cancel the common factor.

$\frac{1}{4 \cdot 2} (4 (19 π))$

Step 21.13.4

Rewrite the expression.

$\frac{1}{2} (19 π)$

Step 21.14

Combine $19$ and $\frac{1}{2}$ .

$\frac{19}{2} π$

Step 21.15

Combine $\frac{19}{2}$ and $π$ .

$\frac{19 π}{2}$

Step 22

The result can be shown in multiple forms.

Exact Form:

$\frac{19 π}{2}$

Decimal Form:

$29.84513020 \dots$