Calculus Examples

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

Step 1

Factor out $\tan^{2} (x)$ .

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

Step 2

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

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

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $\sec (x)$ .

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 4

Multiply $u^{2} (- 1 + u^{2})$ .

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

Step 5

Simplify.

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Step 5.1

Move $- 1$ to the left of $u^{2}$ .

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

Step 5.2

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

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

Step 5.3

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 5.3.1

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

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

Step 5.3.2

Add $2$ and $2$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

$- (\frac{1}{3} u^{3} + C) + \int u^{4} d u$

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Simplify.

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

Step 11

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

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