Calculus Examples

$\cos^{5} (x)$

Step 1

Write $\cos^{5} (x)$ as a function.

$f (x) = \cos^{5} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \cos^{5} (x) d x$

Step 4

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

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

Step 5

Simplify with factoring out.

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Step 5.1

Factor $2$ out of $4$ .

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

Step 5.2

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

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

Step 6

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

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

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $\sin (x)$ .

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

Step 7.1.2

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

$\cos (x)$

Step 7.2

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Apply the distributive property.

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

Step 8.5

Move $u^{2}$ .

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

Step 8.6

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 8.7

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 8.8

Multiply $- 1$ by $1$ .

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

Step 8.9

Multiply $- 1$ by $1$ .

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

Step 8.10

Multiply $- 1$ by $- 1$ .

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

Step 8.11

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

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

Step 8.12

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 8.13

Add $2$ and $2$ .

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

Step 8.14

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

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

Step 8.15

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

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

Step 8.16

Move $1$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

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

Step 11

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 12

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 13

Apply the constant rule.

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Simplify.

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

Step 15

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

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

Step 16

Reorder terms.

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

Step 17

The answer is the antiderivative of the function $f (x) = \cos^{5} (x)$ .

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