Calculus Examples

$\int \frac{\sec (x)}{\cot (x)} d x$

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\int \sec (x) \tan (x) d x$

Step 2

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$\sec (x) + C$