Calculus Examples

$\int_{0}^{\frac{π}{2}} \sin^{2} (2 x) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

$u_{lower} = 2 \cdot 0$

Step 1.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = 2 \frac{π}{2}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 1.5.2

Rewrite the expression.

$u_{upper} = π$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = π$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{1}{2} \int_{0}^{π} \frac{1 - \cos (2 u_{1})}{2} d u_{1}$

Step 5

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

$\frac{1}{2} (\frac{1}{2} \int_{0}^{π} 1 - \cos (2 u_{1}) d u_{1})$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{2 \cdot 2} \int_{0}^{π} 1 - \cos (2 u_{1}) d u_{1}$

Step 6.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int_{0}^{π} 1 - \cos (2 u_{1}) d u_{1}$

Step 7

Split the single integral into multiple integrals.

$\frac{1}{4} (\int_{0}^{π} d u_{1} + \int_{0}^{π} - \cos (2 u_{1}) d u_{1})$

Step 8

Apply the constant rule.

$\frac{1}{4} (u_{1}]_{0}^{π} + \int_{0}^{π} - \cos (2 u_{1}) d u_{1})$

Step 9

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

$\frac{1}{4} (u_{1}]_{0}^{π} - \int_{0}^{π} \cos (2 u_{1}) d u_{1})$

Step 10

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 10.1

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

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Step 10.1.1

Differentiate $2 u_{1}$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

$u_{lower} = 2 \cdot 0$

Step 10.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 10.4

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

$u_{upper} = 2 π$

Step 10.5

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

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 10.6

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

$\frac{1}{4} (u_{1}]_{0}^{π} - \int_{0}^{2 π} \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 11

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

$\frac{1}{4} (u_{1}]_{0}^{π} - \int_{0}^{2 π} \frac{\cos (u_{2})}{2} d u_{2})$

Step 12

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

$\frac{1}{4} (u_{1}]_{0}^{π} - (\frac{1}{2} \int_{0}^{2 π} \cos (u_{2}) d u_{2}))$

Step 13

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

$\frac{1}{4} (u_{1}]_{0}^{π} - \frac{1}{2} \sin (u_{2})]_{0}^{2 π})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $u_{1}$ at $π$ and at $0$ .

$\frac{1}{4} ((π) + 0 - \frac{1}{2} \sin (u_{2})]_{0}^{2 π})$

Step 14.2

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

$\frac{1}{4} ((π) + 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0)))$

Step 14.3

Add $π$ and $0$ .

$\frac{1}{4} (π - \frac{1}{2} (\sin (2 π) - \sin (0)))$

Step 15

Simplify.

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Step 15.1

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

$\frac{1}{4} (π - \frac{1}{2} (\sin (2 π) - 0))$

Step 15.2

Multiply $- 1$ by $0$ .

$\frac{1}{4} (π - \frac{1}{2} (\sin (2 π) + 0))$

Step 15.3

Add $\sin (2 π)$ and $0$ .

$\frac{1}{4} (π - \frac{1}{2} \sin (2 π))$

Step 15.4

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

$\frac{1}{4} (π - \frac{\sin (2 π)}{2})$

Step 16

Simplify.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

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

$\frac{1}{4} (π - \frac{\sin (0)}{2})$

Step 16.1.2

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

$\frac{1}{4} (π - \frac{0}{2})$

Step 16.2

Divide $0$ by $2$ .

$\frac{1}{4} (π - 0)$

Step 16.3

Multiply $- 1$ by $0$ .

$\frac{1}{4} (π + 0)$

Step 16.4

Add $π$ and $0$ .

$\frac{1}{4} π$

Step 16.5

Combine $\frac{1}{4}$ and $π$ .

$\frac{π}{4}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{4}$

Decimal Form:

$0.78539816 \dots$