Calculus Examples

$\int \cos^{7} (x) d x$

Step 1

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

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $6$ .

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

Step 2.2

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

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

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

Step 3

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

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

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 $d u$ .

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

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

Step 5

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

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Step 5.1

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 5.2

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 5.3

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 5.4

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 5.5

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 5.6

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 5.7

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 5.8

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 5.9

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 5.10

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 5.11

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 5.12

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 5.13

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 5.14

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 5.15

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 5.16

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 5.17

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 5.18

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 5.19

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 5.20

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 5.21

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 5.22

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 5.23

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 5.24

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 5.25

Multiply $- 1$ by $- 1$ .

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

Step 5.26

Multiply $- 1$ by $1$ .

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

Step 5.27

Factor out negative.

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

Step 5.28

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 5.29

Add $2$ and $2$ .

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

Step 5.30

Factor out negative.

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

Step 5.31

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 5.32

Add $4$ and $2$ .

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

Step 5.33

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

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

Step 5.34

Move $1$ .

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

Step 5.35

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

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

Step 5.36

Move $1$ .

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

Step 5.37

Move $- 3 u^{2}$ .

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

Step 5.38

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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 14

Simplify.

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Step 14.1

Simplify.

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Step 14.1.1

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 14.1.2

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 14.1.3

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 14.2

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 15

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

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

Step 16

Reorder terms.

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

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