Find the Antiderivative f(x)=2e^x-3xe^x

$f (x) = 2 e^{x} - 3 x e^{x}$

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.

$F (x) = \int 2 e^{x} - 3 x e^{x} d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 e^{x} d x + \int - 3 x e^{x} d x$

Step 4

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

$2 \int e^{x} d x + \int - 3 x e^{x} d x$

Step 5

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

$2 (e^{x} + C) + \int - 3 x e^{x} d x$

Step 6

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$2 (e^{x} + C) - 3 \int x e^{x} d x$

Step 7

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

$2 (e^{x} + C) - 3 (x e^{x} - \int e^{x} d x)$

Step 8

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

$2 (e^{x} + C) - 3 (x e^{x} - (e^{x} + C))$

Step 9

Simplify.

$2 e^{x} - 3 x e^{x} + 3 e^{x} + C$

Add $2 e^{x}$ and $3 e^{x}$ .

$- 3 x e^{x} + 5 e^{x} + C$

Step 10

The answer is the antiderivative of the function $f (x) = 2 e^{x} - 3 x e^{x}$ .

$F (x) =$ $- 3 x e^{x} + 5 e^{x} + C$