Calculus Examples

$\int_{1}^{8} (1 - x) e^{- x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 1 - x$ and $d v = e^{- x}$ .

$(1 - x) (- e^{- x})]_{1}^{8} - \int_{1}^{8} - e^{- x} \cdot - 1 d x$

Step 2

Simplify.

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Step 2.1

Multiply $- 1$ by $- 1$ .

$(1 - x) (- e^{- x})]_{1}^{8} - \int_{1}^{8} 1 e^{- x} d x$

Step 2.2

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

$(1 - x) (- e^{- x})]_{1}^{8} - \int_{1}^{8} e^{- x} d x$

Step 3

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

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

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Step 3.1.1

Differentiate $- x$ .

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

Step 3.1.2

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 3.1.3

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 3.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 3.2

Substitute the lower limit in for $x$ in $u = - x$ .

$u_{lower} = - 1 \cdot 1$

Step 3.3

Multiply $- 1$ by $1$ .

$u_{lower} = - 1$

Step 3.4

Substitute the upper limit in for $x$ in $u = - x$ .

$u_{upper} = - 1 \cdot 8$

Step 3.5

Multiply $- 1$ by $8$ .

$u_{upper} = - 8$

Step 3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = - 1$

$u_{upper} = - 8$

Step 3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$(1 - x) (- e^{- x})]_{1}^{8} - \int_{- 1}^{- 8} - e^{u} d u$

Step 4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$(1 - x) (- e^{- x})]_{1}^{8} - - \int_{- 1}^{- 8} e^{u} d u$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$(1 - x) (- e^{- x})]_{1}^{8} + 1 \int_{- 1}^{- 8} e^{u} d u$

Step 5.2

Multiply $\int_{- 1}^{- 8} e^{u} d u$ by $1$ .

$(1 - x) (- e^{- x})]_{1}^{8} + \int_{- 1}^{- 8} e^{u} d u$

Step 6

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

$(1 - x) (- e^{- x})]_{1}^{8} + e^{u}]_{- 1}^{- 8}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $(1 - x) (- e^{- x})$ at $8$ and at $1$ .

$((1 - 1 \cdot 8) (- e^{- 1 \cdot 8})) - (1 - 1 \cdot 1) (- e^{- 1 \cdot 1}) + e^{u}]_{- 1}^{- 8}$

Step 7.2

Evaluate $e^{u}$ at $- 8$ and at $- 1$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 1$ by $8$ .

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

Step 7.3.2

Subtract $8$ from $1$ .

$- 7 (- e^{- 1 \cdot 8}) - (1 - 1 \cdot 1) (- e^{- 1 \cdot 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.3

Multiply $- 1$ by $8$ .

$- 7 (- e^{- 8}) - (1 - 1 \cdot 1) (- e^{- 1 \cdot 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.4

Multiply $- 1$ by $- 7$ .

$7 e^{- 8} - (1 - 1 \cdot 1) (- e^{- 1 \cdot 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.5

Multiply $- 1$ by $1$ .

$7 e^{- 8} - (1 - 1) (- e^{- 1 \cdot 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.6

Subtract $1$ from $1$ .

$7 e^{- 8} - 0 (- e^{- 1 \cdot 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.7

Multiply $- 1$ by $0$ .

$7 e^{- 8} + 0 (- e^{- 1 \cdot 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.8

Multiply $- 1$ by $1$ .

$7 e^{- 8} + 0 (- e^{- 1}) + (e^{- 8}) - e^{- 1}$

Step 7.3.9

Multiply $- 1$ by $0$ .

$7 e^{- 8} + 0 e^{- 1} + (e^{- 8}) - e^{- 1}$

Step 7.3.10

Multiply $0$ by $e^{- 1}$ .

$7 e^{- 8} + 0 + (e^{- 8}) - e^{- 1}$

Step 7.3.11

Add $7 e^{- 8}$ and $0$ .

$7 e^{- 8} + (e^{- 8}) - e^{- 1}$

Step 7.3.12

Add $7 e^{- 8}$ and $e^{- 8}$ .

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

Step 8

Simplify each term.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.36519574 \dots$

Step 10