Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

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

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $- x$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2

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

$u_{lower} = - 0$

Step 4.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 4.4

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

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

Step 4.5

Multiply $- 1$ by $1$ .

$u_{upper} = - 1$

Step 4.6

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

$u_{lower} = 0$

$u_{upper} = - 1$

Step 4.7

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

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

Step 5

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

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

Evaluate $e^{u}$ at $- 1$ and at $0$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 1$ by $1$ .

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

Step 7.3.2

Multiply $- 1$ by $1$ .

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

Step 7.3.3

Multiply $- 1$ by $0$ .

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

Step 7.3.4

Anything raised to $0$ is $1$ .

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

Step 7.3.5

Multiply $- 1$ by $1$ .

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

Step 7.3.6

Multiply $0$ by $- 1$ .

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

Step 7.3.7

Add $- e^{- 1}$ and $0$ .

$- e^{- 1} - ((e^{- 1}) - e^{0})$

Step 7.3.8

Anything raised to $0$ is $1$ .

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

Step 7.3.9

Multiply $- 1$ by $1$ .

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

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.1.3

Apply the distributive property.

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

Step 8.1.4

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Combine the numerators over the common denominator.

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

Step 8.3

Subtract $1$ from $- 1$ .

$1 + \frac{- 2}{e}$

Step 8.4

Move the negative in front of the fraction.

$1 - \frac{2}{e}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$1 - \frac{2}{e}$

Decimal Form:

$0.26424111 \dots$

Step 10