Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

$\frac{1}{3} x^{3} + C + \int \sin (x) d x$

Step 5

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

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

Step 6

Simplify.

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

Step 7

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

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