Calculus Examples

$\int \sin^{3} (x) \cos^{5} (x) d x$

Step 1

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

$\int \sin^{3} (x) (\cos^{4} (x) \cos (x)) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $4$ .

$\int \sin^{3} (x) ({\cos (x)}^{2 (2)} \cos (x)) d x$

Step 2.2

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

$\int \sin^{3} (x) ({(\cos^{2} (x))}^{2} \cos (x)) d x$

$\int \sin^{3} (x) ({(\cos^{2} (x))}^{2} \cos (x)) d x$

Step 3

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

$\int \sin^{3} (x) ({(1 - \sin^{2} (x))}^{2} \cos (x)) d x$

Step 4

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

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Step 4.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 4.1.2

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

$\cos (x)$

$\cos (x)$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int u^{3} {(1 - u^{2})}^{2} d u$

$\int u^{3} {(1 - u^{2})}^{2} d u$

Step 5

Expand $u^{3} {(1 - u^{2})}^{2}$ .

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Step 5.1

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

$\int u^{3} ((1 - u^{2}) (1 - u^{2})) d u$

Step 5.2

Apply the distributive property.

$\int u^{3} (1 (1 - u^{2}) - u^{2} (1 - u^{2})) d u$

Step 5.3

Apply the distributive property.

$\int u^{3} (1 \cdot 1 + 1 (- u^{2}) - u^{2} (1 - u^{2})) d u$

Step 5.4

Apply the distributive property.

$\int u^{3} (1 \cdot 1 + 1 (- u^{2}) - u^{2} \cdot 1 - u^{2} (- u^{2})) d u$

Step 5.5

Apply the distributive property.

$\int u^{3} (1 \cdot 1 + 1 (- u^{2})) + u^{3} (- u^{2} \cdot 1 - u^{2} (- u^{2})) d u$

Step 5.6

Apply the distributive property.

$\int u^{3} (1 \cdot 1) + u^{3} (1 (- u^{2})) + u^{3} (- u^{2} \cdot 1 - u^{2} (- u^{2})) d u$

Step 5.7

Apply the distributive property.

$\int u^{3} (1 \cdot 1) + u^{3} (1 (- u^{2})) + u^{3} (- u^{2} \cdot 1) + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.8

Move $u^{3}$ .

$\int 1 \cdot 1 u^{3} + u^{3} (1 (- u^{2})) + u^{3} (- u^{2} \cdot 1) + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.9

Move parentheses.

$\int 1 \cdot 1 u^{3} + u^{3} (1 \cdot - 1) u^{2} + u^{3} (- u^{2} \cdot 1) + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.10

Move $u^{3}$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} + u^{3} (- u^{2} \cdot 1) + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.11

Move $u^{2}$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} + u^{3} (- 1 \cdot 1 u^{2}) + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.12

Move parentheses.

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} + u^{3} (- 1 \cdot 1) u^{2} + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.13

Move $u^{3}$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} + u^{3} (- u^{2} (- u^{2})) d u$

Step 5.14

Move $u^{2}$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} + u^{3} (- 1 \cdot - 1 u^{2} u^{2}) d u$

Step 5.15

Move parentheses.

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} + u^{3} (- 1 \cdot - 1 u^{2}) u^{2} d u$

Step 5.16

Move parentheses.

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} + u^{3} (- 1 \cdot - 1) u^{2} u^{2} d u$

Step 5.17

Move $u^{3}$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.18

Multiply $1$ by $1$ .

$\int 1 u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.19

Multiply $u^{3}$ by $1$ .

$\int u^{3} + 1 \cdot - 1 u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.20

Multiply $- 1$ by $1$ .

$\int u^{3} - u^{3} u^{2} - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.21

Factor out negative.

$\int u^{3} - (u^{3} u^{2}) - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.22

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

$\int u^{3} - u^{3 + 2} - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.23

Add $3$ and $2$ .

$\int u^{3} - u^{5} - 1 \cdot 1 u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.24

Multiply $- 1$ by $1$ .

$\int u^{3} - u^{5} - u^{3} u^{2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.25

Factor out negative.

$\int u^{3} - u^{5} - (u^{3} u^{2}) - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.26

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

$\int u^{3} - u^{5} - u^{3 + 2} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.27

Add $3$ and $2$ .

$\int u^{3} - u^{5} - u^{5} - 1 \cdot - 1 u^{3} u^{2} u^{2} d u$

Step 5.28

Multiply $- 1$ by $- 1$ .

$\int u^{3} - u^{5} - u^{5} + 1 u^{3} u^{2} u^{2} d u$

Step 5.29

Multiply $u^{3}$ by $1$ .

$\int u^{3} - u^{5} - u^{5} + u^{3} u^{2} u^{2} d u$

Step 5.30

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

$\int u^{3} - u^{5} - u^{5} + u^{3 + 2} u^{2} d u$

Step 5.31

Add $3$ and $2$ .

$\int u^{3} - u^{5} - u^{5} + u^{5} u^{2} d u$

Step 5.32

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

$\int u^{3} - u^{5} - u^{5} + u^{5 + 2} d u$

Step 5.33

Add $5$ and $2$ .

$\int u^{3} - u^{5} - u^{5} + u^{7} d u$

Step 5.34

Subtract $u^{5}$ from $- u^{5}$ .

$\int u^{3} - 2 u^{5} + u^{7} d u$

Step 5.35

Reorder $- 2 u^{5}$ and $u^{7}$ .

$\int u^{3} + u^{7} - 2 u^{5} d u$

Step 5.36

Move $u^{3}$ .

$\int u^{7} - 2 u^{5} + u^{3} d u$

$\int u^{7} - 2 u^{5} + u^{3} d u$

Step 6

Split the single integral into multiple integrals.

$\int u^{7} d u + \int - 2 u^{5} d u + \int u^{3} d u$

Step 7

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

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

Step 8

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

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

Step 9

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{8} u^{8} + C - 2 (\frac{1}{6} u^{6} + C) + \int u^{3} d u$

Step 10

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{8} u^{8} + C - 2 (\frac{1}{6} u^{6} + C) + \frac{1}{4} u^{4} + C$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{6}$ and $u^{6}$ .

$\frac{1}{8} u^{8} + C - 2 (\frac{u^{6}}{6} + C) + \frac{1}{4} u^{4} + C$

Step 11.2

Simplify.

$\frac{u^{8}}{8} - \frac{u^{6}}{3} + \frac{1}{4} u^{4} + C$

$\frac{u^{8}}{8} - \frac{u^{6}}{3} + \frac{1}{4} u^{4} + C$

Step 12

Replace all occurrences of $u$ with $\sin (x)$ .

$\frac{\sin^{8} (x)}{8} - \frac{\sin^{6} (x)}{3} + \frac{1}{4} \sin^{4} (x) + C$

Step 13

Reorder terms.

$\frac{1}{8} \sin^{8} (x) - \frac{1}{3} \sin^{6} (x) + \frac{1}{4} \sin^{4} (x) + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$