Calculus Examples

$\int 8 \cos (θ) \cos^{5} (\sin (θ)) d θ$

Step 1

Since $8$ is constant with respect to $θ$ , move $8$ out of the integral.

$8 \int \cos (θ) \cos^{5} (\sin (θ)) d θ$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $\sin (θ)$ .

$\frac{d}{d θ} [\sin (θ)]$

Step 2.1.2

The derivative of $\sin (θ)$ with respect to $θ$ is $\cos (θ)$ .

$\cos (θ)$

Step 2.2

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

$8 \int \cos^{5} (u_{1}) d u_{1}$

Step 3

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

$8 \int \cos^{4} (u_{1}) \cos (u_{1}) d u_{1}$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $2$ out of $4$ .

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

Step 4.2

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

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

Step 5

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

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

Step 6

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 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

$\cos (u_{1})$

Step 6.2

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 7.4

Apply the distributive property.

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

Step 7.5

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

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

Step 7.6

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

$8 \int 1 \cdot 1 + 1 \cdot - 1 {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 7.7

Multiply $1$ by $1$ .

$8 \int 1 + 1 \cdot - 1 {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 7.8

Multiply $- 1$ by $1$ .

$8 \int 1 - {u_{2}}^{2} - 1 \cdot 1 {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 7.9

Multiply $- 1$ by $1$ .

$8 \int 1 - {u_{2}}^{2} - {u_{2}}^{2} - 1 \cdot - 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 7.10

Multiply $- 1$ by $- 1$ .

$8 \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + 1 {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 7.11

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

$8 \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + {u_{2}}^{2} {u_{2}}^{2} d u_{2}$

Step 7.12

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

$8 \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + {u_{2}}^{2 + 2} d u_{2}$

Step 7.13

Add $2$ and $2$ .

$8 \int 1 - {u_{2}}^{2} - {u_{2}}^{2} + {u_{2}}^{4} d u_{2}$

Step 7.14

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

$8 \int 1 - 2 {u_{2}}^{2} + {u_{2}}^{4} d u_{2}$

Step 7.15

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

$8 \int 1 + {u_{2}}^{4} - 2 {u_{2}}^{2} d u_{2}$

Step 7.16

Move $1$ .

$8 \int {u_{2}}^{4} - 2 {u_{2}}^{2} + 1 d u_{2}$

Step 8

Split the single integral into multiple integrals.

$8 (\int {u_{2}}^{4} d u_{2} + \int - 2 {u_{2}}^{2} d u_{2} + \int d u_{2})$

Step 9

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

$8 (\frac{1}{5} {u_{2}}^{5} + C + \int - 2 {u_{2}}^{2} d u_{2} + \int d u_{2})$

Step 10

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

$8 (\frac{1}{5} {u_{2}}^{5} + C - 2 \int {u_{2}}^{2} d u_{2} + \int d u_{2})$

Step 11

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

$8 (\frac{1}{5} {u_{2}}^{5} + C - 2 (\frac{1}{3} {u_{2}}^{3} + C) + \int d u_{2})$

Step 12

Apply the constant rule.

$8 (\frac{1}{5} {u_{2}}^{5} + C - 2 (\frac{1}{3} {u_{2}}^{3} + C) + u_{2} + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

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

$8 (\frac{1}{5} {u_{2}}^{5} + C - 2 (\frac{{u_{2}}^{3}}{3} + C) + u_{2} + C)$

Step 13.1.2

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

$8 (\frac{{u_{2}}^{5}}{5} + C - 2 (\frac{{u_{2}}^{3}}{3} + C) + u_{2} + C)$

Step 13.2

Simplify.

$8 (\frac{{u_{2}}^{5}}{5} - \frac{2 {u_{2}}^{3}}{3} + u_{2}) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

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

$8 (\frac{\sin^{5} (u_{1})}{5} - \frac{2 \sin^{3} (u_{1})}{3} + \sin (u_{1})) + C$

Step 14.2

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

$8 (\frac{\sin^{5} (\sin (θ))}{5} - \frac{2 \sin^{3} (\sin (θ))}{3} + \sin (\sin (θ))) + C$

Step 15

Reorder terms.

$8 (\frac{1}{5} \sin^{5} (\sin (θ)) - \frac{2}{3} \sin^{3} (\sin (θ)) + \sin (\sin (θ))) + C$