Calculus Examples

$x^{3} e^{x}$

Step 1

Write $x^{3} e^{x}$ as a function.

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

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.

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

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{3}$ and $d v = e^{x}$ .

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

Step 5

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

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

Step 6

Multiply $3$ by $- 1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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