Calculus Examples

$f (x) = \frac{9}{\cos^{2} (x)} - 3 \sin (x)$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \frac{9}{\cos^{2} (x)} - 3 \sin (x) d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{9}{\cos^{2} (x)} d x + \int - 3 \sin (x) d x$

Step 4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int \frac{1}{\cos^{2} (x)} d x + \int - 3 \sin (x) d x$

Step 5

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$9 \int \sec^{2} (x) d x + \int - 3 \sin (x) d x$

Step 6

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$9 (\tan (x) + C) + \int - 3 \sin (x) d x$

Step 7

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

$9 (\tan (x) + C) - 3 \int \sin (x) d x$

Step 8

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

$9 (\tan (x) + C) - 3 (- \cos (x) + C)$

Step 9

Simplify.

$9 \tan (x) + 3 \cos (x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{9}{\cos^{2} (x)} - 3 \sin (x)$ .

$F (x) =$ $9 \tan (x) + 3 \cos (x) + C$