Calculus Examples

$f (x) = \frac{1}{4} \cdot \sin^{2} (2 x)$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \frac{1}{4} \cdot \sin^{2} (2 x) d x$

Step 3

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

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $2 x$ .

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

Step 4.1.2

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 4.1.3

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

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $4$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

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

Step 10.2

Multiply $2$ by $8$ .

$\frac{1}{16} \int 1 - \cos (2 u_{1}) d u_{1}$

Step 11

Split the single integral into multiple integrals.

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

Step 12

Apply the constant rule.

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

Step 13

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

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

Step 14

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 14.1

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

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Step 14.1.1

Differentiate $2 u_{1}$ .

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

Step 14.1.2

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 14.1.3

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 14.1.4

Multiply $2$ by $1$ .

$2$

Step 14.2

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

$\frac{1}{16} (u_{1} + C - \frac{1}{2} (\sin (u_{2}) + C))$

Step 18

Simplify.

$\frac{1}{16} (u_{1} - \frac{1}{2} \sin (u_{2})) + C$

Step 19

Substitute back in for each integration substitution variable.

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Step 19.1

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

$\frac{1}{16} (2 x - \frac{1}{2} \sin (u_{2})) + C$

Step 19.2

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

$\frac{1}{16} (2 x - \frac{1}{2} \sin (2 u_{1})) + C$

Step 19.3

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

$\frac{1}{16} (2 x - \frac{1}{2} \sin (2 (2 x))) + C$

Step 20

Simplify.

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Step 20.1

Simplify each term.

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Step 20.1.1

Multiply $2$ by $2$ .

$\frac{1}{16} (2 x - \frac{1}{2} \sin (4 x)) + C$

Step 20.1.2

Combine $\sin (4 x)$ and $\frac{1}{2}$ .

$\frac{1}{16} (2 x - \frac{\sin (4 x)}{2}) + C$

Step 20.2

Apply the distributive property.

$\frac{1}{16} (2 x) + \frac{1}{16} (- \frac{\sin (4 x)}{2}) + C$

Step 20.3

Cancel the common factor of $2$ .

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Step 20.3.1

Factor $2$ out of $16$ .

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

Step 20.3.2

Factor $2$ out of $2 x$ .

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

Step 20.3.3

Cancel the common factor.

$\frac{1}{2 \cdot 8} (2 x) + \frac{1}{16} (- \frac{\sin (4 x)}{2}) + C$

Step 20.3.4

Rewrite the expression.

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

Step 20.4

Combine $\frac{1}{8}$ and $x$ .

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

Step 20.5

Multiply $\frac{1}{16} (- \frac{\sin (4 x)}{2})$ .

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Step 20.5.1

Multiply $\frac{1}{16}$ by $\frac{\sin (4 x)}{2}$ .

$\frac{x}{8} - \frac{\sin (4 x)}{16 \cdot 2} + C$

Step 20.5.2

Multiply $16$ by $2$ .

$\frac{x}{8} - \frac{\sin (4 x)}{32} + C$

Step 21

Reorder terms.

$\frac{1}{8} x - \frac{1}{32} \sin (4 x) + C$

Step 22

The answer is the antiderivative of the function $f (x) = \frac{1}{4} \cdot \sin^{2} (2 x)$ .

$F (x) =$ $\frac{1}{8} x - \frac{1}{32} \sin (4 x) + C$