Calculus Examples

\int {sin}^{5} (x) d x

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

Step 1

Factor out

{sin}^{4} (x)

$\sin^{4} (x)$ .

\int {sin}^{4} (x) sin (x) d x

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 2.2

Rewrite

{sin (x)}^{2 (2)}

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

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

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

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

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

Step 3

Using the Pythagorean Identity, rewrite

{sin}^{2} (x)

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

1 - {cos}^{2} (x)

$1 - \cos^{2} (x)$ .

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

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

Step 4

Let

u = cos (x)

$u = \cos (x)$ . Then

d u = - sin (x) d x

$d u = - \sin (x) d x$ , so

- \frac{1}{sin (x)} d u = d x

$- \frac{1}{\sin (x)} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 4.1

Let

u = cos (x)

$u = \cos (x)$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate

cos (x)

$\cos (x)$ .

\frac{d}{d x} [cos (x)]

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

Step 4.1.2

The derivative of

cos (x)

$\cos (x)$ with respect to

x

$x$ is

- sin (x)

$- \sin (x)$ .

- sin (x)

$- \sin (x)$

- sin (x)

$- \sin (x)$

Step 4.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 5

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

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

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

Step 6

Expand

{(1 - u^{2})}^{2}

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

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Step 6.1

Rewrite

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

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Move

$u^{2}$ .

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

Step 6.6

Move

$u^{2}$ .

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

Step 6.7

Multiply

$1$ by

$1$ .

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

Step 6.8

Multiply

$- 1$ by

$1$ .

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

Step 6.9

Multiply

$- 1$ by

$1$ .

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

Step 6.10

Multiply

$- 1$ by

$- 1$ .

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

Step 6.11

Multiply

$u^{2}$ by

$1$ .

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

Step 6.12

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.13

Add

$2$ and

$2$ .

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

Step 6.14

Subtract

$u^{2}$ from

$- u^{2}$ .

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

Step 6.15

Reorder

$- 2 u^{2}$ and

$u^{4}$ .

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

Step 6.16

Move

$1$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

By the Power Rule, the integral of

$u^{4}$ with respect to

$u$ is

$\frac{1}{5} u^{5}$ .

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

Step 9

Since

$- 2$ is constant with respect to

$u$ , move

$- 2$ out of the integral.

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

Step 10

By the Power Rule, the integral of

$u^{2}$ with respect to

$u$ is

$\frac{1}{3} u^{3}$ .

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

Step 11

Apply the constant rule.

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

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

Combine

$\frac{1}{3}$ and

$u^{3}$ .

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

Step 12.1.2

Combine

$\frac{1}{5}$ and

$u^{5}$ .

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

Step 12.2

Simplify.

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

Step 13

Replace all occurrences of

$u$ with

$\cos (x)$ .

$- (\frac{\cos^{5} (x)}{5} - \frac{2 \cos^{3} (x)}{3} + \cos (x)) + C$

Step 14

Reorder terms.

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