Calculus Examples

$\int_{- 8}^{0} x e^{x} d x$

Step 1

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

$x e^{x}]_{- 8}^{0} - \int_{- 8}^{0} e^{x} d x$

Step 2

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

$x e^{x}]_{- 8}^{0} - (e^{x}]_{- 8}^{0})$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $x e^{x}$ at $0$ and at $- 8$ .

$(0 e^{0}) + 8 e^{- 8} - (e^{x}]_{- 8}^{0})$

Step 3.2

Evaluate $e^{x}$ at $0$ and at $- 8$ .

$(0 e^{0}) + 8 e^{- 8} - ((e^{0}) - e^{- 8})$

Step 3.3

Simplify.

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Step 3.3.1

Anything raised to $0$ is $1$ .

$0 \cdot 1 + 8 e^{- 8} - ((e^{0}) - e^{- 8})$

Step 3.3.2

Multiply $0$ by $1$ .

$0 + 8 e^{- 8} - ((e^{0}) - e^{- 8})$

Step 3.3.3

Add $0$ and $8 e^{- 8}$ .

$8 e^{- 8} - ((e^{0}) - e^{- 8})$

Step 3.3.4

Anything raised to $0$ is $1$ .

$8 e^{- 8} - (1 - e^{- 8})$

Step 4

Simplify each term.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 \frac{1}{e^{8}} - (1 - e^{- 8})$

Step 4.2

Combine $8$ and $\frac{1}{e^{8}}$ .

$\frac{8}{e^{8}} - (1 - e^{- 8})$

Step 4.3

Apply the distributive property.

$\frac{8}{e^{8}} - 1 \cdot 1 - - e^{- 8}$

Step 4.4

Multiply $- 1$ by $1$ .

$\frac{8}{e^{8}} - 1 - - e^{- 8}$

Step 4.5

Multiply $- - e^{- 8}$ .

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

$\frac{8}{e^{8}} - 1 + 1 e^{- 8}$

Step 4.5.2

Multiply $e^{- 8}$ by $1$ .

$\frac{8}{e^{8}} - 1 + e^{- 8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{e^{8}} - 1 + e^{- 8}$

Decimal Form:

$- 0.99698083 \dots$

Step 6