Calculus Examples

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

Step 1

Write $\frac{1}{\cos^{4} (x)}$ as a function.

$f (x) = \frac{1}{\cos^{4} (x)}$

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.

$F (x) = \int \frac{1}{\cos^{4} (x)} d x$

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $\tan (x)$ .

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

Step 7.1.2

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

$\sec^{2} (x)$

Step 7.2

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

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

Apply the constant rule.

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

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

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

Step 13

The answer is the antiderivative of the function $f (x) = \frac{1}{\cos^{4} (x)}$ .

$F (x) =$ $\tan (x) + \frac{1}{3} \tan^{3} (x) + C$