Calculus Examples

$x + e^{x}$

Step 1

Write $x + e^{x}$ as a function.

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

Step 2

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 3

Set up the integral to solve.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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

Step 8

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

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