Calculus Examples

$\frac{3}{\sec (x)}$

Step 1

Write $\frac{3}{\sec (x)}$ as a function.

$f (x) = \frac{3}{\sec (x)}$

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.

$F (x) = \int \frac{3}{\sec (x)} d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{1}{\sec (x)} d x$

Step 5

Simplify.

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Step 5.1

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

$3 \int \frac{1}{\frac{1}{\cos (x)}} d x$

Step 5.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$3 \int 1 \cos (x) d x$

Step 5.3

Multiply $\cos (x)$ by $1$ .

$3 \int \cos (x) d x$

Step 6

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$3 (\sin (x) + C)$

Step 7

Simplify.

$3 \sin (x) + C$

Step 8

The answer is the antiderivative of the function $f (x) = \frac{3}{\sec (x)}$ .

$F (x) =$ $3 \sin (x) + C$