Calculus Examples

$\frac{\sin (x)}{\cos^{5} (x)}$

Step 1

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

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

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.

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

Step 4

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- d u = \sin (x) d x$ . 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 $- \sin (x)$ .

$- \sin (x)$

Step 4.2

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

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Apply basic rules of exponents.

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Step 7.1

Move $u^{5}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.2

Multiply the exponents in ${(u^{5})}^{- 1}$ .

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Step 7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2.2

Multiply $5$ by $- 1$ .

$- \int u^{- 5} d u$

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $u^{- 4}$ and $\frac{1}{4}$ .

$- (- \frac{u^{- 4}}{4} + C)$

Step 9.1.2

Move $u^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9.2

Simplify.

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

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{4 u^{4}} + C$

Step 9.3.2

Multiply $\frac{1}{4 u^{4}}$ by $1$ .

$\frac{1}{4 u^{4}} + C$

Step 10

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

$\frac{1}{4 \cos^{4} (x)} + C$

Step 11

Simplify.

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Step 11.1

Multiply by $1$ .

$\frac{1}{4 (\cos^{4} (x) \cdot 1)} + C$

Step 11.2

Separate fractions.

$\frac{1}{4 \cdot (1)} \cdot \frac{1}{\cos^{4} (x)} + C$

Step 11.3

Convert from $\frac{1}{\cos^{4} (x)}$ to $\sec^{4} (x)$ .

$\frac{1}{4 \cdot (1)} \sec^{4} (x) + C$

Step 11.4

Multiply $4$ by $1$ .

$\frac{1}{4} \sec^{4} (x) + C$

Step 11.5

Combine $\frac{1}{4}$ and $\sec^{4} (x)$ .

$\frac{\sec^{4} (x)}{4} + C$

Step 11.6

Reorder terms.

$\frac{1}{4} \sec^{4} (x) + C$

Step 12

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

$F (x) =$ $\frac{1}{4} \sec^{4} (x) + C$