Calculus Examples

$\int_{0}^{8} x e^{1 - x} d x$

Step 1

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

$x (- e^{1 - x})]_{0}^{8} - \int_{0}^{8} - e^{1 - x} d x$

Step 2

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

$x (- e^{1 - x})]_{0}^{8} - - \int_{0}^{8} e^{1 - x} d x$

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

$x (- e^{1 - x})]_{0}^{8} + 1 \int_{0}^{8} e^{1 - x} d x$

Step 3.2

Multiply $\int_{0}^{8} e^{1 - x} d x$ by $1$ .

$x (- e^{1 - x})]_{0}^{8} + \int_{0}^{8} e^{1 - x} d x$

Step 4

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 1 - x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $1 - x$ .

$\frac{d}{d x} [1 - x]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x]$

Step 4.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x]$

Step 4.1.3

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

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Step 4.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - 1 \cdot 1$

Step 4.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 4.1.4

Subtract $1$ from $0$ .

$- 1$

Step 4.2

Substitute the lower limit in for $x$ in $u = 1 - x$ .

$u_{lower} = 1 - 0$

Step 4.3

Subtract $0$ from $1$ .

$u_{lower} = 1$

Step 4.4

Substitute the upper limit in for $x$ in $u = 1 - x$ .

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

Step 4.5

Simplify.

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Step 4.5.1

Multiply $- 1$ by $8$ .

$u_{upper} = 1 - 8$

Step 4.5.2

Subtract $8$ from $1$ .

$u_{upper} = - 7$

Step 4.6

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

$u_{lower} = 1$

$u_{upper} = - 7$

Step 4.7

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

$x (- e^{1 - x})]_{0}^{8} + \int_{1}^{- 7} - e^{u} d u$

Step 5

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

$x (- e^{1 - x})]_{0}^{8} - \int_{1}^{- 7} e^{u} d u$

Step 6

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

$x (- e^{1 - x})]_{0}^{8} - (e^{u}]_{1}^{- 7})$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $x (- e^{1 - x})$ at $8$ and at $0$ .

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

Step 7.2

Evaluate $e^{u}$ at $- 7$ and at $1$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 1$ by $8$ .

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

Step 7.3.2

Subtract $8$ from $1$ .

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

Step 7.3.3

Multiply $- 1$ by $8$ .

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

Step 7.3.4

Multiply $- 1$ by $0$ .

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

Step 7.3.5

Add $1$ and $0$ .

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

Step 7.3.6

Simplify.

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

Step 7.3.7

Multiply $- 1$ by $0$ .

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

Step 7.3.8

Multiply $0$ by $e$ .

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

Step 7.3.9

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

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

Step 7.3.10

Simplify.

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

Step 8

Simplify each term.

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Step 8.1

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

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

Step 8.2

Combine $- 8$ and $\frac{1}{e^{7}}$ .

$\frac{- 8}{e^{7}} - (e^{- 7} - e)$

Step 8.3

Move the negative in front of the fraction.

$- \frac{8}{e^{7}} - (e^{- 7} - e)$

Step 8.4

Apply the distributive property.

$- \frac{8}{e^{7}} - e^{- 7} - - e$

Step 8.5

Multiply $- - e$ .

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Step 8.5.1

Multiply $- 1$ by $- 1$ .

$- \frac{8}{e^{7}} - e^{- 7} + 1 e$

Step 8.5.2

Multiply $e$ by $1$ .

$- \frac{8}{e^{7}} - e^{- 7} + e$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{e^{7}} - e^{- 7} + e$

Decimal Form:

$2.71007489 \dots$

Step 10