Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

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

Step 1.2

Separate fractions.

$\int \frac{1}{\cos^{2} (x)} \cdot \frac{\tan (x)}{\cos (x)} d x$

Step 1.3

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 1.4

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

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

Step 1.5

Simplify.

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Step 1.5.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\int \frac{1}{\cos^{2} (x)} (\tan (x) \frac{1}{\cos (x)}) d x$

Step 1.5.2

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

$\int \frac{1}{\cos^{2} (x)} (\tan (x) \sec (x)) d x$

Step 1.6

Rewrite $1$ as $1^{2}$ .

$\int \frac{1^{2}}{\cos^{2} (x)} (\tan (x) \sec (x)) d x$

Step 1.7

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$\int {(\frac{1}{\cos (x)})}^{2} (\tan (x) \sec (x)) d x$

Step 1.8

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

$\int \sec^{2} (x) (\tan (x) \sec (x)) d x$

Step 1.9

Multiply $\sec^{2} (x)$ by $\sec (x)$ by adding the exponents.

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Step 1.9.1

Move $\sec (x)$ .

$\int \sec (x) \sec^{2} (x) \tan (x) d x$

Step 1.9.2

Multiply $\sec (x)$ by $\sec^{2} (x)$ .

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Step 1.9.2.1

Raise $\sec (x)$ to the power of $1$ .

$\int \sec^{1} (x) \sec^{2} (x) \tan (x) d x$

Step 1.9.2.2

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

$\int {\sec (x)}^{1 + 2} \tan (x) d x$

Step 1.9.3

Add $1$ and $2$ .

$\int \sec^{3} (x) \tan (x) d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $\sec (x)$ .

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

Step 2.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 2.2

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

$\int u^{2} d u$

Step 3

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

$\frac{1}{3} u^{3} + C$

Step 4

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

$\frac{1}{3} \sec^{3} (x) + C$