Calculus Examples

$f (x) = 2 \sin (x) + x^{2}$

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.

$F (x) = \int 2 \sin (x) + x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 \sin (x) d x + \int x^{2} d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sin (x) d x + \int x^{2} d x$

Step 5

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

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

Step 6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$2 (- \cos (x) + C) + \frac{1}{3} x^{3} + C$

Step 7

Simplify.

$- 2 \cos (x) + \frac{1}{3} x^{3} + C$

Step 8

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

$F (x) =$ $- 2 \cos (x) + \frac{1}{3} x^{3} + C$