Calculus Examples

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

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

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1 - \cos (2 u_{1})}{2}$ by $\frac{1 + \cos (2 u_{1})}{2}$ .

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

Step 6.2

Multiply $2$ by $2$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Multiply $4$ by $2$ .

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

Step 9

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 9.1

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

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Step 9.1.1

Differentiate $2 u_{1}$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

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

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

Step 10

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

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

Step 11

Simplify by multiplying through.

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Step 11.1

Simplify.

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Step 11.1.1

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

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

Step 11.1.2

Multiply $2$ by $8$ .

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

Step 11.2

Expand $(1 - \cos (u_{2})) (1 + \cos (u_{2}))$ .

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Step 11.2.1

Apply the distributive property.

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

Step 11.2.2

Apply the distributive property.

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

Step 11.2.3

Apply the distributive property.

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

Step 11.2.4

Move $\cos (u_{2})$ .

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

Step 11.2.5

Multiply $1$ by $1$ .

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

Step 11.2.6

Multiply $\cos (u_{2})$ by $1$ .

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

Step 11.2.7

Multiply $- 1$ by $1$ .

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

Step 11.2.8

Factor out negative.

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

Step 11.2.9

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 11.2.10

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 11.2.11

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

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

Step 11.2.12

Add $1$ and $1$ .

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

Step 11.2.13

Subtract $\cos (u_{2})$ from $\cos (u_{2})$ .

$\frac{1}{16} \int 1 + 0 - \cos^{2} (u_{2}) d u_{2}$

Step 11.2.14

Subtract $\cos^{2} (u_{2})$ from $0$ .

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

Step 12

Split the single integral into multiple integrals.

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

Step 13

Apply the constant rule.

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Split the single integral into multiple integrals.

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

Step 18

Apply the constant rule.

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

Step 19

Let $u_{3} = 2 u_{2}$ . Then $d u_{3} = 2 d u_{2}$ , so $\frac{1}{2} d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 19.1

Let $u_{3} = 2 u_{2}$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 19.1.1

Differentiate $2 u_{2}$ .

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

Step 19.1.2

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

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

Step 19.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 19.1.4

Multiply $2$ by $1$ .

$2$

Step 19.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\frac{1}{16} (u_{2} + C - \frac{1}{2} (u_{2} + C + \int \cos (u_{3}) \frac{1}{2} d u_{3}))$

Step 20

Combine $\cos (u_{3})$ and $\frac{1}{2}$ .

$\frac{1}{16} (u_{2} + C - \frac{1}{2} (u_{2} + C + \int \frac{\cos (u_{3})}{2} d u_{3}))$

Step 21

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{16} (u_{2} + C - \frac{1}{2} (u_{2} + C + \frac{1}{2} \int \cos (u_{3}) d u_{3}))$

Step 22

The integral of $\cos (u_{3})$ with respect to $u_{3}$ is $\sin (u_{3})$ .

$\frac{1}{16} (u_{2} + C - \frac{1}{2} (u_{2} + C + \frac{1}{2} (\sin (u_{3}) + C)))$

Step 23

Simplify.

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Step 23.1

Simplify.

$\frac{1}{16} (u_{2} - \frac{u_{2}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 23.2

Simplify.

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Step 23.2.1

To write $u_{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{16} (u_{2} \cdot \frac{2}{2} - \frac{u_{2}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 23.2.2

Combine $u_{2}$ and $\frac{2}{2}$ .

$\frac{1}{16} (\frac{u_{2} \cdot 2}{2} - \frac{u_{2}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 23.2.3

Combine the numerators over the common denominator.

$\frac{1}{16} (\frac{u_{2} \cdot 2 - u_{2}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 23.2.4

Move $2$ to the left of $u_{2}$ .

$\frac{1}{16} (\frac{2 \cdot u_{2} - u_{2}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 23.2.5

Subtract $u_{2}$ from $2 u_{2}$ .

$\frac{1}{16} (\frac{u_{2}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 24

Substitute back in for each integration substitution variable.

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Step 24.1

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

$\frac{1}{16} (\frac{2 u_{1}}{2} - \frac{\sin (u_{3})}{4}) + C$

Step 24.2

Replace all occurrences of $u_{3}$ with $2 u_{2}$ .

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

Step 24.3

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

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

Step 24.4

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

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

Step 24.5

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

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

Step 25

Simplify.

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Step 25.1

Simplify each term.

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Step 25.1.1

Cancel the common factor of $2$ .

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Step 25.1.1.1

Cancel the common factor.

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

Step 25.1.1.2

Divide $2 x$ by $1$ .

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

Step 25.1.2

Multiply $2 \cdot 2 \cdot 2$ .

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Step 25.1.2.1

Multiply $2$ by $2$ .

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

Step 25.1.2.2

Multiply $4$ by $2$ .

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

Step 25.2

Apply the distributive property.

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

Step 25.3

Cancel the common factor of $2$ .

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Step 25.3.1

Factor $2$ out of $16$ .

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

Step 25.3.2

Factor $2$ out of $2 x$ .

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

Step 25.3.3

Cancel the common factor.

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

Step 25.3.4

Rewrite the expression.

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

Step 25.4

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

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

Step 25.5

Multiply $\frac{1}{16} (- \frac{\sin (8 x)}{4})$ .

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Step 25.5.1

Multiply $\frac{1}{16}$ by $\frac{\sin (8 x)}{4}$ .

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

Step 25.5.2

Multiply $16$ by $4$ .

$\frac{x}{8} - \frac{\sin (8 x)}{64} + C$

Step 26

Reorder terms.

$\frac{1}{8} x - \frac{1}{64} \sin (8 x) + C$