Calculus Examples

$\int \sec^{4} (x) \tan^{3} (x) d x$

Step 1

Simplify the expression.

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Step 1.1

Rewrite $4$ as $2$ plus $2$

$\int {\sec (x)}^{2 + 2} \tan^{3} (x) d x$

Step 1.2

Rewrite ${\sec (x)}^{2 + 2}$ as $\sec^{2} (x) \sec^{2} (x)$ .

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

Step 2

Using the Pythagorean Identity, rewrite $\sec^{2} (x)$ as $1 + \tan^{2} (x)$ .

$\int (1 + \tan^{2} (x)) \sec^{2} (x) \tan^{3} (x) d x$

Step 3

Let $u = \tan (x)$ . Then $d u = \sec^{2} (x) d x$ , so $\frac{1}{\sec^{2} (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

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

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Step 3.1.1

Differentiate $\tan (x)$ .

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

Step 3.1.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 3.2

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

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $u^{3}$ by $1$ .

$\int u^{3} + u^{2} u^{3} d u$

Step 4.3

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

$\int u^{3} + u^{2 + 3} d u$

Step 4.4

Add $2$ and $3$ .

$\int u^{3} + u^{5} d u$

Step 4.5

Reorder $u^{3}$ and $u^{5}$ .

$\int u^{5} + u^{3} d u$

Step 5

Split the single integral into multiple integrals.

$\int u^{5} d u + \int u^{3} d u$

Step 6

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{6} u^{6} + C + \int u^{3} d u$

Step 7

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{6} u^{6} + C + \frac{1}{4} u^{4} + C$

Step 8

Simplify.

$\frac{1}{6} u^{6} + \frac{1}{4} u^{4} + C$

Step 9

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

$\frac{1}{6} \tan^{6} (x) + \frac{1}{4} \tan^{4} (x) + C$