Calculus Examples

$f (x) = e^{x} + x$

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.

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

Step 3

Split the single integral into multiple integrals.

$\int e^{x} d x + \int x d x$

Step 4

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

$e^{x} + C + \int x d x$

Step 5

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

$e^{x} + C + \frac{1}{2} x^{2} + C$

Step 6

Simplify.

$e^{x} + \frac{1}{2} x^{2} + C$

Step 7

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

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