Calculus Examples

$\sin^{7} (x)$

Step 1

Factor out $\sin^{6} (x)$ .

$\int \sin^{6} (x) \sin (x) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $6$ .

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

Step 2.2

Rewrite ${\sin (x)}^{2 (3)}$ as exponentiation.

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\cos (x)$ .

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

Step 4.1.2

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

$- \sin (x)$

$- \sin (x)$

Step 4.2

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

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

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

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 6

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

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Step 6.1

Use the Binomial Theorem.

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

Step 6.2

Rewrite the exponentiation as a product.

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

Step 6.3

Rewrite the exponentiation as a product.

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

Step 6.4

Rewrite the exponentiation as a product.

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

Step 6.5

Rewrite the exponentiation as a product.

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

Step 6.6

Rewrite the exponentiation as a product.

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

Step 6.7

Rewrite the exponentiation as a product.

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

Step 6.8

Move $u^{2}$ .

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

Step 6.9

Move parentheses.

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

Step 6.10

Move parentheses.

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

Step 6.11

Move $u^{2}$ .

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

Step 6.12

Move parentheses.

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

Step 6.13

Move parentheses.

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

Step 6.14

Move $u^{2}$ .

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

Step 6.15

Multiply $1$ by $1$ .

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

Step 6.16

Multiply $1$ by $1$ .

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

Step 6.17

Multiply $3$ by $1$ .

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

Step 6.18

Multiply $3$ by $1$ .

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

Step 6.19

Multiply $3$ by $- 1$ .

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

Step 6.20

Multiply $3$ by $1$ .

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

Step 6.21

Multiply $3$ by $- 1$ .

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

Step 6.22

Multiply $- 3$ by $- 1$ .

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

Step 6.23

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

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

Step 6.24

Add $2$ and $2$ .

$- \int 1 - 3 u^{2} + 3 u^{4} - 1 \cdot - 1 \cdot - 1 u^{2} u^{2} u^{2} d u$

Step 6.25

Multiply $- 1$ by $- 1$ .

$- \int 1 - 3 u^{2} + 3 u^{4} + 1 \cdot - 1 u^{2} u^{2} u^{2} d u$

Step 6.26

Multiply $- 1$ by $1$ .

$- \int 1 - 3 u^{2} + 3 u^{4} - u^{2} u^{2} u^{2} d u$

Step 6.27

Factor out negative.

$- \int 1 - 3 u^{2} + 3 u^{4} - (u^{2} u^{2}) u^{2} d u$

Step 6.28

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

$- \int 1 - 3 u^{2} + 3 u^{4} - u^{2 + 2} u^{2} d u$

Step 6.29

Add $2$ and $2$ .

$- \int 1 - 3 u^{2} + 3 u^{4} - u^{4} u^{2} d u$

Step 6.30

Factor out negative.

$- \int 1 - 3 u^{2} + 3 u^{4} - (u^{4} u^{2}) d u$

Step 6.31

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

$- \int 1 - 3 u^{2} + 3 u^{4} - u^{4 + 2} d u$

Step 6.32

Add $4$ and $2$ .

$- \int 1 - 3 u^{2} + 3 u^{4} - u^{6} d u$

Step 6.33

Reorder $1$ and $- 3 u^{2}$ .

$- \int - 3 u^{2} + 1 + 3 u^{4} - u^{6} d u$

Step 6.34

Move $1$ .

$- \int - 3 u^{2} + 3 u^{4} + 1 - u^{6} d u$

Step 6.35

Reorder $- 3 u^{2}$ and $3 u^{4}$ .

$- \int 3 u^{4} - 3 u^{2} + 1 - u^{6} d u$

Step 6.36

Move $1$ .

$- \int 3 u^{4} - 3 u^{2} - u^{6} + 1 d u$

Step 6.37

Move $- 3 u^{2}$ .

$- \int 3 u^{4} - u^{6} - 3 u^{2} + 1 d u$

Step 6.38

Reorder $3 u^{4}$ and $- u^{6}$ .

$- \int - u^{6} + 3 u^{4} - 3 u^{2} + 1 d u$

$- \int - u^{6} + 3 u^{4} - 3 u^{2} + 1 d u$

Step 7

Split the single integral into multiple integrals.

$- (\int - u^{6} d u + \int 3 u^{4} d u + \int - 3 u^{2} d u + \int d u)$

Step 8

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- (- \int u^{6} d u + \int 3 u^{4} d u + \int - 3 u^{2} d u + \int d u)$

Step 9

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$- (- (\frac{1}{7} u^{7} + C) + \int 3 u^{4} d u + \int - 3 u^{2} d u + \int d u)$

Step 10

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$- (- (\frac{1}{7} u^{7} + C) + 3 \int u^{4} d u + \int - 3 u^{2} d u + \int d u)$

Step 11

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

$- (- (\frac{1}{7} u^{7} + C) + 3 (\frac{1}{5} u^{5} + C) + \int - 3 u^{2} d u + \int d u)$

Step 12

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

$- (- (\frac{1}{7} u^{7} + C) + 3 (\frac{1}{5} u^{5} + C) - 3 \int u^{2} d u + \int d u)$

Step 13

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

$- (- (\frac{1}{7} u^{7} + C) + 3 (\frac{1}{5} u^{5} + C) - 3 (\frac{1}{3} u^{3} + C) + \int d u)$

Step 14

Apply the constant rule.

$- (- (\frac{1}{7} u^{7} + C) + 3 (\frac{1}{5} u^{5} + C) - 3 (\frac{1}{3} u^{3} + C) + u + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

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Step 15.1.1

Combine $\frac{1}{7}$ and $u^{7}$ .

$- (- (\frac{u^{7}}{7} + C) + 3 (\frac{1}{5} u^{5} + C) - 3 (\frac{1}{3} u^{3} + C) + u + C)$

Step 15.1.2

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

$- (- (\frac{u^{7}}{7} + C) + 3 (\frac{u^{5}}{5} + C) - 3 (\frac{1}{3} u^{3} + C) + u + C)$

Step 15.1.3

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

$- (- (\frac{u^{7}}{7} + C) + 3 (\frac{u^{5}}{5} + C) - 3 (\frac{u^{3}}{3} + C) + u + C)$

$- (- (\frac{u^{7}}{7} + C) + 3 (\frac{u^{5}}{5} + C) - 3 (\frac{u^{3}}{3} + C) + u + C)$

Step 15.2

Simplify.

$- (- \frac{u^{7}}{7} + \frac{3 u^{5}}{5} - u^{3} + u) + C$

$- (- \frac{u^{7}}{7} + \frac{3 u^{5}}{5} - u^{3} + u) + C$

Step 16

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

$- (- \frac{\cos^{7} (x)}{7} + \frac{3 \cos^{5} (x)}{5} - \cos^{3} (x) + \cos (x)) + C$

Step 17

Reorder terms.

$- (- \frac{1}{7} \cos^{7} (x) + \frac{3}{5} \cos^{5} (x) - \cos^{3} (x) + \cos (x)) + C$

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