Calculus Examples

$x e^{x} d x$

Step 1

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

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

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.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Raise $x$ to the power of $1$ .

$\int x^{1} x e^{x} d d x$

Step 4.2

Raise $x$ to the power of $1$ .

$\int x^{1} x^{1} e^{x} d d x$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int x^{1 + 1} e^{x} d d x$

Step 4.4

Add $1$ and $1$ .

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

Step 5

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

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

Step 6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{x}$ .

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

Step 7

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

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

Step 8

Multiply $2$ by $- 1$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Rewrite $d (x^{2} e^{x} - 2 (x e^{x} - (e^{x} + C)))$ as $d (x^{2} e^{x} - 2 x e^{x} + 2 e^{x}) + C$ .

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

Step 12

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

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