Calculus Examples

$\int \cos^{5} (11 x) d x$

Step 1

Let $u_{1} = 11 x$ . Then $d u_{1} = 11 d x$ , so $\frac{1}{11} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 11 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $11 x$ .

$\frac{d}{d x} [11 x]$

Step 1.1.2

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$11 \frac{d}{d x} [x]$

Step 1.1.3

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

$11 \cdot 1$

Step 1.1.4

Multiply $11$ by $1$ .

$11$

Step 1.2

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

$\int \cos^{5} (u_{1}) \frac{1}{11} d u_{1}$

Step 2

Combine $\cos^{5} (u_{1})$ and $\frac{1}{11}$ .

$\int \frac{\cos^{5} (u_{1})}{11} d u_{1}$

Step 3

Since $\frac{1}{11}$ is constant with respect to $u_{1}$ , move $\frac{1}{11}$ out of the integral.

$\frac{1}{11} \int \cos^{5} (u_{1}) d u_{1}$

Step 4

Factor out $\cos^{4} (u_{1})$ .

$\frac{1}{11} \int \cos^{4} (u_{1}) \cos (u_{1}) d u_{1}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $2$ out of $4$ .

$\frac{1}{11} \int {\cos (u_{1})}^{2 (2)} \cos (u_{1}) d u_{1}$

Step 5.2

Rewrite ${\cos (u_{1})}^{2 (2)}$ as exponentiation.

$\frac{1}{11} \int {(\cos^{2} (u_{1}))}^{2} \cos (u_{1}) d u_{1}$

Step 6

Using the Pythagorean Identity, rewrite $\cos^{2} (u_{1})$ as $1 - \sin^{2} (u_{1})$ .

$\frac{1}{11} \int {(1 - \sin^{2} (u_{1}))}^{2} \cos (u_{1}) d u_{1}$

Step 7

Let $u_{2} = \sin (u_{1})$ . Then $d u_{2} = \cos (u_{1}) d u_{1}$ , so $\frac{1}{\cos (u_{1})} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.1

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

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Step 7.1.1

Differentiate $\sin (u_{1})$ .

$\frac{d}{d u_{1}} [\sin (u_{1})]$

Step 7.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1})$

Step 7.2

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

$\frac{1}{11} \int {(1 - {u_{2}}^{2})}^{2} d u_{2}$

Step 8

Expand ${(1 - {u_{2}}^{2})}^{2}$ .

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Step 8.1

Rewrite ${(1 - {u_{2}}^{2})}^{2}$ as $(1 - {u_{2}}^{2}) (1 - {u_{2}}^{2})$ .

$\frac{1}{11} \int (1 - {u_{2}}^{2}) (1 - {u_{2}}^{2}) d u_{2}$

Step 8.2

Apply the distributive property.

$\frac{1}{11} \int 1 (1 - {u_{2}}^{2}) - {u_{2}}^{2} (1 - {u_{2}}^{2}) d u_{2}$

Step 8.3

Apply the distributive property.

$\frac{1}{11} \int 1 \cdot 1 + 1 (- {u_{2}}^{2}) - {u_{2}}^{2} (1 - {u_{2}}^{2}) d u_{2}$

Step 8.4

Apply the distributive property.

$\frac{1}{11} \int 1 \cdot 1 + 1 (- {u_{2}}^{2}) - {u_{2}}^{2} \cdot 1 - {u_{2}}^{2} (- {u_{2}}^{2}) d u_{2}$

Step 8.5

Move ${u_{2}}^{2}$ .

$\frac{1}{11} \int 1 \cdot 1 + 1 \cdot - 1 {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - {u_{2}}^{2} (- {u_{2}}^{2}) d u_{2}$

Step 8.6

Move ${u_{2}}^{2}$ .

$\frac{1}{11} \int 1 \cdot 1 + 1 \cdot - 1 {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.7

Multiply $1$ by $1$ .

$\frac{1}{11} \int 1 + 1 \cdot - 1 {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.8

Multiply $- 1$ by $1$ .

$\frac{1}{11} \int 1 - {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.9

Multiply $- 1$ by $1$ .

$\frac{1}{11} \int 1 - {u_{2}}^{2} - {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.10

Multiply $- 1$ by $- 1$ .

$\frac{1}{11} \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.11

Multiply ${u_{2}}^{2}$ by $1$ .

$\frac{1}{11} \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 8.12

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

$\frac{1}{11} \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + {u_{2}}^{2 + 2} d u_{2}$

Step 8.13

Add $2$ and $2$ .

$\frac{1}{11} \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + {u_{2}}^{4} d u_{2}$

Step 8.14

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

$\frac{1}{11} \int 1 - 2 {u_{2}}^{2} + {u_{2}}^{4} d u_{2}$

Step 8.15

Reorder $- 2 {u_{2}}^{2}$ and ${u_{2}}^{4}$ .

$\frac{1}{11} \int 1 + {u_{2}}^{4} - 2 {u_{2}}^{2} d u_{2}$

Step 8.16

Move $1$ .

$\frac{1}{11} \int {u_{2}}^{4} - 2 {u_{2}}^{2} + 1 d u_{2}$

Step 9

Split the single integral into multiple integrals.

$\frac{1}{11} (\int {u_{2}}^{4} d u_{2} + \int - 2 {u_{2}}^{2} d u_{2} + \int d u_{2})$

Step 10

By the Power Rule, the integral of ${u_{2}}^{4}$ with respect to $u_{2}$ is $\frac{1}{5} {u_{2}}^{5}$ .

$\frac{1}{11} (\frac{1}{5} {u_{2}}^{5} + C + \int - 2 {u_{2}}^{2} d u_{2} + \int d u_{2})$

Step 11

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

$\frac{1}{11} (\frac{1}{5} {u_{2}}^{5} + C - 2 \int {u_{2}}^{2} d u_{2} + \int d u_{2})$

Step 12

By the Power Rule, the integral of ${u_{2}}^{2}$ with respect to $u_{2}$ is $\frac{1}{3} {u_{2}}^{3}$ .

$\frac{1}{11} (\frac{1}{5} {u_{2}}^{5} + C - 2 (\frac{1}{3} {u_{2}}^{3} + C) + \int d u_{2})$

Step 13

Apply the constant rule.

$\frac{1}{11} (\frac{1}{5} {u_{2}}^{5} + C - 2 (\frac{1}{3} {u_{2}}^{3} + C) + u_{2} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

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Step 14.1.1

Combine $\frac{1}{3}$ and ${u_{2}}^{3}$ .

$\frac{1}{11} (\frac{1}{5} {u_{2}}^{5} + C - 2 (\frac{{u_{2}}^{3}}{3} + C) + u_{2} + C)$

Step 14.1.2

Combine $\frac{1}{5}$ and ${u_{2}}^{5}$ .

$\frac{1}{11} (\frac{{u_{2}}^{5}}{5} + C - 2 (\frac{{u_{2}}^{3}}{3} + C) + u_{2} + C)$

Step 14.2

Simplify.

$\frac{1}{11} (\frac{{u_{2}}^{5}}{5} - \frac{2 {u_{2}}^{3}}{3} + u_{2}) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{2}$ with $\sin (u_{1})$ .

$\frac{1}{11} (\frac{\sin^{5} (u_{1})}{5} - \frac{2 \sin^{3} (u_{1})}{3} + \sin (u_{1})) + C$

Step 15.2

Replace all occurrences of $u_{1}$ with $11 x$ .

$\frac{1}{11} (\frac{\sin^{5} (11 x)}{5} - \frac{2 \sin^{3} (11 x)}{3} + \sin (11 x)) + C$

Step 16

Reorder terms.

$\frac{1}{11} (\frac{1}{5} \sin^{5} (11 x) - \frac{2}{3} \sin^{3} (11 x) + \sin (11 x)) + C$