Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 5

Cancel the common factor of $\cos (x)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 7

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$- \cot (x) + C$