Calculus Examples

$\sin^{2} (\frac{π}{4} x)$

Step 1

Write $\sin^{2} (\frac{π}{4} x)$ as a function.

$f (x) = \sin^{2} (\frac{π}{4} x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sin^{2} (\frac{π}{4} x) d x$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\frac{π}{4} x$ .

$\frac{d}{d x} [\frac{π}{4} x]$

Step 4.1.2

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

$\frac{π}{4} \frac{d}{d x} [x]$

Step 4.1.3

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

$\frac{π}{4} \cdot 1$

Step 4.1.4

Multiply $\frac{π}{4}$ by $1$ .

$\frac{π}{4}$

Step 4.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \sin^{2} (u_{1}) \frac{1}{\frac{π}{4}} d u_{1}$

Step 5

Simplify.

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Step 5.1

Multiply by the reciprocal of the fraction to divide by $\frac{π}{4}$ .

$\int \sin^{2} (u_{1}) (1 \frac{4}{π}) d u_{1}$

Step 5.2

Multiply $\frac{4}{π}$ by $1$ .

$\int \sin^{2} (u_{1}) \frac{4}{π} d u_{1}$

Step 5.3

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

$\int \frac{\sin^{2} (u_{1}) \cdot 4}{π} d u_{1}$

Step 5.4

Move $4$ to the left of $\sin^{2} (u_{1})$ .

$\int \frac{4 \sin^{2} (u_{1})}{π} d u_{1}$

Step 6

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

$\frac{4}{π} \int \sin^{2} (u_{1}) d u_{1}$

Step 7

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

$\frac{4}{π} \int \frac{1 - \cos (2 u_{1})}{2} d u_{1}$

Step 8

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

$\frac{4}{π} (\frac{1}{2} \int 1 - \cos (2 u_{1}) d u_{1})$

Step 9

Simplify.

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Step 9.1

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

$\frac{4}{2 π} \int 1 - \cos (2 u_{1}) d u_{1}$

Step 9.2

Cancel the common factor of $4$ and $2$ .

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Step 9.2.1

Factor $2$ out of $4$ .

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

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $2 π$ .

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

Step 9.2.2.2

Cancel the common factor.

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

Step 9.2.2.3

Rewrite the expression.

$\frac{2}{π} \int 1 - \cos (2 u_{1}) d u_{1}$

Step 10

Split the single integral into multiple integrals.

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

Step 11

Apply the constant rule.

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

Step 12

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

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

Step 13

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 13.1

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

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Step 13.1.1

Differentiate $2 u_{1}$ .

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

Step 13.1.2

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 13.1.3

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 13.1.4

Multiply $2$ by $1$ .

$2$

Step 13.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 14

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

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

Step 15

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

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

Step 16

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

$\frac{2}{π} (u_{1} + C - \frac{1}{2} (\sin (u_{2}) + C))$

Step 17

Simplify.

$\frac{2}{π} (u_{1} - \frac{1}{2} \sin (u_{2})) + C$

Step 18

Substitute back in for each integration substitution variable.

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Step 18.1

Replace all occurrences of $u_{1}$ with $\frac{π}{4} x$ .

$\frac{2}{π} (\frac{π}{4} x - \frac{1}{2} \sin (u_{2})) + C$

Step 18.2

Replace all occurrences of $u_{2}$ with $2 u_{1}$ .

$\frac{2}{π} (\frac{π}{4} x - \frac{1}{2} \sin (2 u_{1})) + C$

Step 18.3

Replace all occurrences of $u_{1}$ with $\frac{π}{4} x$ .

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

Step 19

Simplify.

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Step 19.1

Simplify each term.

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Step 19.1.1

Combine $\frac{π}{4}$ and $x$ .

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

Step 19.1.2

Combine $\frac{π}{4}$ and $x$ .

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

Step 19.1.3

Cancel the common factor of $2$ .

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Step 19.1.3.1

Factor $2$ out of $4$ .

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

Step 19.1.3.2

Cancel the common factor.

$\frac{2}{π} (\frac{π x}{4} - \frac{1}{2} \sin (2 \frac{π x}{2 \cdot 2})) + C$

Step 19.1.3.3

Rewrite the expression.

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

Step 19.1.4

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

$\frac{2}{π} (\frac{π x}{4} - \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.2

Apply the distributive property.

$\frac{2}{π} \cdot \frac{π x}{4} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.3

Cancel the common factor of $2$ .

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Step 19.3.1

Factor $2$ out of $4$ .

$\frac{2}{π} \cdot \frac{π x}{2 (2)} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.3.2

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π x}{2 \cdot 2} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.3.3

Rewrite the expression.

$\frac{1}{π} \cdot \frac{π x}{2} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.4

Cancel the common factor of $π$ .

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Step 19.4.1

Factor $π$ out of $π x$ .

$\frac{1}{π} \cdot \frac{π (x)}{2} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.4.2

Cancel the common factor.

$\frac{1}{π} \cdot \frac{π x}{2} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.4.3

Rewrite the expression.

$\frac{x}{2} + \frac{2}{π} (- \frac{\sin (\frac{π x}{2})}{2}) + C$

Step 19.5

Cancel the common factor of $2$ .

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Step 19.5.1

Move the leading negative in $- \frac{\sin (\frac{π x}{2})}{2}$ into the numerator.

$\frac{x}{2} + \frac{2}{π} \cdot \frac{- \sin (\frac{π x}{2})}{2} + C$

Step 19.5.2

Cancel the common factor.

$\frac{x}{2} + \frac{2}{π} \cdot \frac{- \sin (\frac{π x}{2})}{2} + C$

Step 19.5.3

Rewrite the expression.

$\frac{x}{2} + \frac{1}{π} (- \sin (\frac{π x}{2})) + C$

Step 19.6

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

$\frac{x}{2} - \frac{\sin (\frac{π x}{2})}{π} + C$

Step 20

Reorder terms.

$\frac{1}{2} x - \frac{1}{π} \sin (\frac{π}{2} x) + C$

Step 21

The answer is the antiderivative of the function $f (x) = \sin^{2} (\frac{π}{4} x)$ .

$F (x) =$ $\frac{1}{2} x - \frac{1}{π} \sin (\frac{π}{2} x) + C$