Calculus Examples

$16 \sin^{2} (2 x)$

Step 1

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int \sin^{2} (2 x) d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $2 x$ .

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

Step 2.1.2

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 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

$16 \int \sin^{2} (u_{1}) \frac{1}{2} d u_{1}$

Step 3

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

$16 \int \frac{\sin^{2} (u_{1})}{2} d u_{1}$

Step 4

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

$16 (\frac{1}{2} \int \sin^{2} (u_{1}) d u_{1})$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $16$ .

$\frac{16}{2} \int \sin^{2} (u_{1}) d u_{1}$

Step 5.2

Cancel the common factor of $16$ and $2$ .

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Step 5.2.1

Factor $2$ out of $16$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

Divide $8$ by $1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $8$ .

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

Step 8.2

Cancel the common factor of $8$ and $2$ .

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Step 8.2.1

Factor $2$ out of $8$ .

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

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.2.2

Cancel the common factor.

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

Step 8.2.2.3

Rewrite the expression.

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

Step 8.2.2.4

Divide $4$ by $1$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

$4 (u_{1} + C + \int - \cos (2 u_{1}) d u_{1})$

Step 11

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

$4 (u_{1} + C - \int \cos (2 u_{1}) d u_{1})$

Step 12

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 12.1

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

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Step 12.1.1

Differentiate $2 u_{1}$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

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

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

Step 18

Simplify.

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Step 18.1

Simplify each term.

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Step 18.1.1

Multiply $2$ by $2$ .

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

Step 18.1.2

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

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

Step 18.2

Apply the distributive property.

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

Step 18.3

Multiply $2$ by $4$ .

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

Step 18.4

Cancel the common factor of $2$ .

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Step 18.4.1

Move the leading negative in $- \frac{\sin (4 x)}{2}$ into the numerator.

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

Step 18.4.2

Factor $2$ out of $4$ .

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

Step 18.4.3

Cancel the common factor.

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

Step 18.4.4

Rewrite the expression.

$8 x + 2 (- \sin (4 x)) + C$

Step 18.5

Multiply $- 1$ by $2$ .

$8 x - 2 \sin (4 x) + C$