Calculus Examples

$- 13 \sin (x) + \frac{5}{\cos^{2} (x)}$

Step 1

Write $- 13 \sin (x) + \frac{5}{\cos^{2} (x)}$ as a function.

$f (x) = - 13 \sin (x) + \frac{5}{\cos^{2} (x)}$

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.

$F (x) = \int - 13 \sin (x) + \frac{5}{\cos^{2} (x)} d x$

Step 4

Split the single integral into multiple integrals.

$\int - 13 \sin (x) d x + \int \frac{5}{\cos^{2} (x)} d x$

Step 5

Since $- 13$ is constant with respect to $x$ , move $- 13$ out of the integral.

$- 13 \int \sin (x) d x + \int \frac{5}{\cos^{2} (x)} d x$

Step 6

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

$- 13 (- \cos (x) + C) + \int \frac{5}{\cos^{2} (x)} d x$

Step 7

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$- 13 (- \cos (x) + C) + 5 \int \frac{1}{\cos^{2} (x)} d x$

Step 8

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

$- 13 (- \cos (x) + C) + 5 \int \sec^{2} (x) d x$

Step 9

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

$- 13 (- \cos (x) + C) + 5 (\tan (x) + C)$

Step 10

Simplify.

$13 \cos (x) + 5 \tan (x) + C$

Step 11

The answer is the antiderivative of the function $f (x) = - 13 \sin (x) + \frac{5}{\cos^{2} (x)}$ .

$F (x) =$ $13 \cos (x) + 5 \tan (x) + C$