Calculus Examples

$f (x) = 9 e^{x} + 6 \sec^{2} (x)$

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.

$F (x) = \int 9 e^{x} + 6 \sec^{2} (x) d x$

Step 3

Split the single integral into multiple integrals.

$\int 9 e^{x} d x + \int 6 \sec^{2} (x) d x$

Step 4

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

$9 \int e^{x} d x + \int 6 \sec^{2} (x) d x$

Step 5

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$9 (e^{x} + C) + \int 6 \sec^{2} (x) d x$

Step 6

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$9 (e^{x} + C) + 6 \int \sec^{2} (x) d x$

Step 7

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

$9 (e^{x} + C) + 6 (\tan (x) + C)$

Step 8

Simplify.

$9 e^{x} + 6 \tan (x) + C$

Step 9

The answer is the antiderivative of the function $f (x) = 9 e^{x} + 6 \sec^{2} (x)$ .

$F (x) =$ $9 e^{x} + 6 \tan (x) + C$