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 $d$ $u$ .

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Step 1.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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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.

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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.

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Step 11.1

Simplify each term.

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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.

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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 $- 1$ .

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

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