Calculus Examples

$\sin (x) + x$

Write $\sin (x) + x$ as a function.

$f (x) = \sin (x) + x$

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Set up the integral to solve.

$F (x) = \int \sin (x) + x d x$

Split the single integral into multiple integrals.

$\int \sin (x) d x + \int x d x$

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

$- \cos (x) + C + \int x d x$

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

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

Simplify.

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

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

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