Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $\tan (x)$ .

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

Step 1.1.2

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

$\sec^{2} (x)$

Step 1.2

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Cancel the common factor of $1$ .

Tap for more steps...

Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Multiply $1 - u^{2}$ by $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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

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

Step 6

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 7

Simplify.

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

Step 8

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

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