Calculus Examples

$\frac{1}{2} \sec (x) \tan (x)$

Step 1

Write $\frac{1}{2} \sec (x) \tan (x)$ as a function.

$f (x) = \frac{1}{2} \sec (x) \tan (x)$

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.

$F (x) = \int \frac{1}{2} \cdot (\sec (x) \tan (x)) d x$

Step 4

Simplify.

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Combine $\sec (x)$ and $\frac{1}{2}$ .

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

Combine $\frac{\sec (x)}{2}$ and $\tan (x)$ .

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

Step 5

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 6

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

$\frac{1}{2} (\sec (x) + C)$

Step 7

Simplify.

$\frac{1}{2} \sec (x) + C$

Step 8

The answer is the antiderivative of the function $f (x) = \frac{1}{2} \cdot (\sec (x) \tan (x))$ .

$F (x) =$ $\frac{1}{2} \sec (x) + C$