Calculus Examples

$x e^{- 5 x}$

Step 1

Write $x e^{- 5 x}$ as a function.

$f (x) = x e^{- 5 x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x e^{- 5 x} d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 5 x}$ .

$x (- \frac{1}{5} e^{- 5 x}) - \int - \frac{1}{5} e^{- 5 x} d x$

Step 5

Simplify.

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Step 5.1

Combine $e^{- 5 x}$ and $\frac{1}{5}$ .

$x (- \frac{e^{- 5 x}}{5}) - \int - \frac{1}{5} e^{- 5 x} d x$

Step 5.2

Combine $x$ and $\frac{e^{- 5 x}}{5}$ .

$- \frac{x e^{- 5 x}}{5} - \int - \frac{1}{5} e^{- 5 x} d x$

Step 6

Since $- \frac{1}{5}$ is constant with respect to $x$ , move $- \frac{1}{5}$ out of the integral.

$- \frac{x e^{- 5 x}}{5} - (- \frac{1}{5} \int e^{- 5 x} d x)$

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $- 1$ .

$- \frac{x e^{- 5 x}}{5} + 1 (\frac{1}{5} \int e^{- 5 x} d x)$

Step 7.2

Multiply $\frac{1}{5}$ by $1$ .

$- \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int e^{- 5 x} d x$

Step 8

Let $u = - 5 x$ . Then $d u = - 5 d x$ , so $- \frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = - 5 x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $- 5 x$ .

$\frac{d}{d x} [- 5 x]$

Step 8.1.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$- 5 \frac{d}{d x} [x]$

Step 8.1.3

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

$- 5 \cdot 1$

Step 8.1.4

Multiply $- 5$ by $1$ .

$- 5$

Step 8.2

Rewrite the problem using $u$ and $d u$ .

$- \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int e^{u} \frac{1}{- 5} d u$

Step 9

Simplify.

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Step 9.1

Move the negative in front of the fraction.

$- \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int e^{u} (- \frac{1}{5}) d u$

Step 9.2

Combine $e^{u}$ and $\frac{1}{5}$ .

$- \frac{x e^{- 5 x}}{5} + \frac{1}{5} \int - \frac{e^{u}}{5} d u$

Step 10

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

$- \frac{x e^{- 5 x}}{5} + \frac{1}{5} (- \int \frac{e^{u}}{5} d u)$

Step 11

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$- \frac{x e^{- 5 x}}{5} + \frac{1}{5} (- (\frac{1}{5} \int e^{u} d u))$

Step 12

Simplify.

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Step 12.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$- \frac{x e^{- 5 x}}{5} - \frac{1}{5 \cdot 5} \int e^{u} d u$

Step 12.2

Multiply $5$ by $5$ .

$- \frac{x e^{- 5 x}}{5} - \frac{1}{25} \int e^{u} d u$

Step 13

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

$- \frac{x e^{- 5 x}}{5} - \frac{1}{25} (e^{u} + C)$

Step 14

Simplify.

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Step 14.1

Rewrite $- \frac{x e^{- 5 x}}{5} - \frac{1}{25} (e^{u} + C)$ as $- \frac{1}{5} x e^{- 5 x} - \frac{1}{25} e^{u} + C$ .

$- \frac{1}{5} x e^{- 5 x} - \frac{1}{25} e^{u} + C$

Step 14.2

Simplify.

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Step 14.2.1

Combine $x$ and $\frac{1}{5}$ .

$- \frac{x}{5} e^{- 5 x} - \frac{1}{25} e^{u} + C$

Step 14.2.2

Combine $e^{- 5 x}$ and $\frac{x}{5}$ .

$- \frac{e^{- 5 x} x}{5} - \frac{1}{25} e^{u} + C$

Step 15

Replace all occurrences of $u$ with $- 5 x$ .

$- \frac{e^{- 5 x} x}{5} - \frac{1}{25} e^{- 5 x} + C$

Step 16

Combine $e^{- 5 x}$ and $\frac{1}{25}$ .

$- \frac{e^{- 5 x} x}{5} - \frac{e^{- 5 x}}{25} + C$

Step 17

Reorder terms.

$- \frac{1}{5} e^{- 5 x} x - \frac{1}{25} e^{- 5 x} + C$

Step 18

The answer is the antiderivative of the function $f (x) = x e^{- 5 x}$ .

$F (x) =$ $- \frac{1}{5} e^{- 5 x} x - \frac{1}{25} e^{- 5 x} + C$