Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

$\int \frac{- 1}{u^{3}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{1}{u^{3}} d u$

Step 3

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

$- \int \frac{1}{u^{3}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $3$ by $- 1$ .

$- \int u^{- 3} d u$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $u^{- 2}$ and $\frac{1}{2}$ .

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

Step 6.1.2

Move $u^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 6.2

Simplify.

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

Step 6.3

Simplify.

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Step 6.3.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{2 u^{2}} + C$

Step 6.3.2

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

$\frac{1}{2 u^{2}} + C$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Multiply by $1$ .

$\frac{1}{2 (\cos^{2} (x) \cdot 1)} + C$

Step 8.2

Separate fractions.

$\frac{1}{2 \cdot (1)} \cdot \frac{1}{\cos^{2} (x)} + C$

Step 8.3

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$\frac{1}{2 \cdot (1)} \sec^{2} (x) + C$

Step 8.4

Multiply $2$ by $1$ .

$\frac{1}{2} \sec^{2} (x) + C$

Step 8.5

Combine $\frac{1}{2}$ and $\sec^{2} (x)$ .

$\frac{\sec^{2} (x)}{2} + C$

Step 8.6

Reorder terms.

$\frac{1}{2} \sec^{2} (x) + C$