Calculus Examples

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

Step 1

Let

$u = - x$ . Then

$d u = - d x$ , so

$- d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

Tap for more steps...

Step 1.1

Let

$u = - x$ . Find

$\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate

$- x$ .

$\frac{d}{d x} [- x]$

Step 1.1.2

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x$ with respect to

$x$ is

$- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$- 1 \cdot 1$

Step 1.1.4

Multiply

$- 1$ by

$1$ .

$- 1$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Factor

$- 1$ out of

$- u$ .

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

Step 2.2

Apply the product rule to

$- (u)$ .

$\int - ({(- 1)}^{2} u^{2}) e^{u} d u$

Step 2.3

Raise

$- 1$ to the power of

$2$ .

$\int - (1 u^{2}) e^{u} d u$

Step 2.4

Multiply

$u^{2}$ by

$1$ .

$\int - u^{2} e^{u} d u$

Step 3

Since

$- 1$ is constant with respect to

$u$ , move

$- 1$ out of the integral.

$- \int u^{2} e^{u} d u$

Step 4

Integrate by parts using the formula

$\int w d v = w v - \int v d w$ , where

$w = u^{2}$ and

$dv = e^{u}$ .

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

Step 5

Since

$2$ is constant with respect to

$u$ , move

$2$ out of the integral.

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

Step 6

Multiply

$2$ by

$- 1$ .

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

Step 7

Integrate by parts using the formula

$\int w d v = w v - \int v d w$ , where

$w = u$ and

$dv = e^{u}$ .

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

Step 8

The integral of

$e^{u}$ with respect to

$u$ is

$e^{u}$ .

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

Step 9

Rewrite

$- (u^{2} e^{u} - 2 (u e^{u} - (e^{u} + C)))$ as

$- (u^{2} e^{u} - 2 u e^{u} + 2 e^{u}) + C$ .

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

Step 10

Replace all occurrences of

$u$ with

$- x$ .

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

Step 11

Simplify.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Apply the product rule to

$- x$ .

$- ({(- 1)}^{2} x^{2} e^{- x} - 2 (- x) e^{- x} + 2 e^{- x}) + C$

Step 11.1.2

Raise

$- 1$ to the power of

$2$ .

$- (1 x^{2} e^{- x} - 2 (- x) e^{- x} + 2 e^{- x}) + C$

Step 11.1.3

Multiply

$x^{2}$ by

$1$ .

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

Step 11.1.4

Multiply

$- 1$ by

$- 2$ .

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Simplify.

Tap for more steps...

Step 11.3.1

Multiply

$2$ by

$- 1$ .

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

Step 11.3.2

Multiply

$2$ by

$- 1$ .

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

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