Calculus Examples

$\frac{(\sec (x))}{\tan (x)}$

Step 1

Write $\frac{(\sec (x))}{\tan (x)}$ as a function.

$f (x) = \frac{(\sec (x))}{\tan (x)}$

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.

$F (x) = \int \frac{\sec (x)}{\tan (x)} d x$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

$\int \csc (x) d x$

Step 9

The integral of $\csc (x)$ with respect to $x$ is $\ln (| \csc (x) - \cot (x) |)$ .

$\ln (| \csc (x) - \cot (x) |) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{\sec (x)}{\tan (x)}$ .

$F (x) =$ $\ln (| \csc (x) - \cot (x) |) + C$