Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Multiply $c$ by $1$ .

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

Step 4

Let $u = - \frac{x}{c}$ . Then $d u = - \frac{1}{c} d x$ , so $- c d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

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

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Step 4.1.1

Differentiate $- \frac{x}{c}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$- \frac{1}{c} \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$- \frac{1}{c}$

Step 4.2

Substitute the lower limit in for $x$ in $u = - \frac{x}{c}$ .

$u_{lower} = - \frac{0}{c}$

Step 4.3

Simplify.

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Step 4.3.1

Divide $0$ by $c$ .

$u_{lower} = - 0$

Step 4.3.2

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $x$ in $u = - \frac{x}{c}$ .

$u_{upper} = - \frac{8}{c}$

Step 4.5

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

$u_{lower} = 0$

$u_{upper} = - \frac{8}{c}$

Step 4.6

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

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

Step 5

Simplify.

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Step 5.1

Dividing two negative values results in a positive value.

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

Step 5.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{c}$ .

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

Step 5.3

Multiply $c$ by $1$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Raise $c$ to the power of $1$ .

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

Step 7.2

Raise $c$ to the power of $1$ .

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

Step 7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.4

Add $1$ and $1$ .

$x (- c e^{- \frac{x}{c}})]_{0}^{8} - c^{2} \int_{0}^{- \frac{8}{c}} e^{u} d u$

Step 8

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

$x (- c e^{- \frac{x}{c}})]_{0}^{8} - c^{2} e^{u}]_{0}^{- \frac{8}{c}}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $x (- c e^{- \frac{x}{c}})$ at $8$ and at $0$ .

$(8 (- c e^{- \frac{8}{c}})) + 0 (- c e^{- \frac{0}{c}}) - c^{2} e^{u}]_{0}^{- \frac{8}{c}}$

Step 9.2

Evaluate $e^{u}$ at $- \frac{8}{c}$ and at $0$ .

$(8 (- c e^{- \frac{8}{c}})) + 0 (- c e^{- \frac{0}{c}}) - c^{2} ((e^{- \frac{8}{c}}) - e^{0})$

Step 9.3

Simplify.

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Step 9.3.1

Multiply $- 1$ by $8$ .

$- 8 (c e^{- \frac{8}{c}}) + 0 (- c e^{- \frac{0}{c}}) - c^{2} ((e^{- \frac{8}{c}}) - e^{0})$

Step 9.3.2

Multiply $- 1$ by $0$ .

$- 8 c e^{- \frac{8}{c}} + 0 (c e^{- \frac{0}{c}}) - c^{2} ((e^{- \frac{8}{c}}) - e^{0})$

Step 9.3.3

Multiply $c$ by $0$ .

$- 8 c e^{- \frac{8}{c}} + 0 e^{- \frac{0}{c}} - c^{2} ((e^{- \frac{8}{c}}) - e^{0})$

Step 9.3.4

Multiply $0$ by $e^{- \frac{0}{c}}$ .

$- 8 c e^{- \frac{8}{c}} + 0 - c^{2} ((e^{- \frac{8}{c}}) - e^{0})$

Step 9.3.5

Add $- 8 c e^{- \frac{8}{c}}$ and $0$ .

$- 8 c e^{- \frac{8}{c}} - c^{2} ((e^{- \frac{8}{c}}) - e^{0})$

Step 9.3.6

Anything raised to $0$ is $1$ .

$- 8 c e^{- \frac{8}{c}} - c^{2} (e^{- \frac{8}{c}} - 1 \cdot 1)$

Step 9.3.7

Multiply $- 1$ by $1$ .

$- 8 c e^{- \frac{8}{c}} - c^{2} (e^{- \frac{8}{c}} - 1)$

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

$- 8 c e^{- \frac{8}{c}} - c^{2} e^{- \frac{8}{c}} - c^{2} \cdot - 1$

Step 10.2

Multiply $- c^{2} \cdot - 1$ .

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Step 10.2.1

Multiply $- 1$ by $- 1$ .

$- 8 c e^{- \frac{8}{c}} - c^{2} e^{- \frac{8}{c}} + 1 c^{2}$

Step 10.2.2

Multiply $c^{2}$ by $1$ .

$- 8 c e^{- \frac{8}{c}} - c^{2} e^{- \frac{8}{c}} + c^{2}$

Step 11