Calculus Examples

$2 x \cdot e^{x}$

Step 1

Write $2 x \cdot e^{x}$ as a function.

$f (x) = 2 x \cdot e^{x}$

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.

$F (x) = \int 2 x \cdot e^{x} d x$

Step 4

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

$2 \int x \cdot e^{x} d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$2 (x e^{x} - \int e^{x} d x)$

Step 6

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

$2 (x e^{x} - (e^{x} + C))$

Step 7

Simplify.

$2 (x e^{x} - e^{x}) + C$

Step 8

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

$F (x) =$ $2 (x e^{x} - e^{x}) + C$