Calculus Examples

$(x + 3) e^{- 2 x}$

Step 1

Write $(x + 3) e^{- 2 x}$ as a function.

$f (x) = (x + 3) e^{- 2 x}$

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.

$F (x) = \int (x + 3) e^{- 2 x} d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x + 3$ and $d v = e^{- 2 x}$ .

$(x + 3) (- \frac{1}{2} e^{- 2 x}) - \int - \frac{1}{2} e^{- 2 x} d x$

Step 5

Since $- \frac{1}{2}$ is constant with respect to $x$ , move $- \frac{1}{2}$ out of the integral.

$(x + 3) (- \frac{1}{2} e^{- 2 x}) - (- \frac{1}{2} \int e^{- 2 x} d x)$

Step 6

Simplify.

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Step 6.1

Combine $e^{- 2 x}$ and $\frac{1}{2}$ .

$(x + 3) (- \frac{e^{- 2 x}}{2}) - (- \frac{1}{2} \int e^{- 2 x} d x)$

Step 6.2

Multiply $- 1$ by $- 1$ .

$(x + 3) (- \frac{e^{- 2 x}}{2}) + 1 (\frac{1}{2} \int e^{- 2 x} d x)$

Step 6.3

Multiply $\frac{1}{2}$ by $1$ .

$(x + 3) (- \frac{e^{- 2 x}}{2}) + \frac{1}{2} \int e^{- 2 x} d x$

Step 7

Let $u = - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1

Differentiate $- 2 x$ .

$\frac{d}{d x} [- 2 x]$

Step 7.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x]$

Step 7.1.3

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

$- 2 \cdot 1$

Step 7.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 7.2

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

$(x + 3) (- \frac{e^{- 2 x}}{2}) + \frac{1}{2} \int e^{u} \frac{1}{- 2} d u$

Step 8

Simplify.

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Step 8.1

Move the negative in front of the fraction.

$(x + 3) (- \frac{e^{- 2 x}}{2}) + \frac{1}{2} \int e^{u} (- \frac{1}{2}) d u$

Step 8.2

Combine $e^{u}$ and $\frac{1}{2}$ .

$(x + 3) (- \frac{e^{- 2 x}}{2}) + \frac{1}{2} \int - \frac{e^{u}}{2} d u$

Step 9

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

$(x + 3) (- \frac{e^{- 2 x}}{2}) + \frac{1}{2} (- \int \frac{e^{u}}{2} d u)$

Step 10

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$(x + 3) (- \frac{e^{- 2 x}}{2}) + \frac{1}{2} (- (\frac{1}{2} \int e^{u} d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$(x + 3) (- \frac{e^{- 2 x}}{2}) - \frac{1}{2 \cdot 2} \int e^{u} d u$

Step 11.2

Multiply $2$ by $2$ .

$(x + 3) (- \frac{e^{- 2 x}}{2}) - \frac{1}{4} \int e^{u} d u$

Step 12

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

$(x + 3) (- \frac{e^{- 2 x}}{2}) - \frac{1}{4} (e^{u} + C)$

Step 13

Simplify.

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Step 13.1

Rewrite $(x + 3) (- \frac{e^{- 2 x}}{2}) - \frac{1}{4} (e^{u} + C)$ as $- x \frac{e^{- 2 x}}{2} + \frac{- 3 e^{- 2 x}}{2} - \frac{1}{4} e^{u} + C$ .

$- x \frac{e^{- 2 x}}{2} + \frac{- 3 e^{- 2 x}}{2} - \frac{1}{4} e^{u} + C$

Step 13.2

Simplify.

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Step 13.2.1

Combine $\frac{e^{- 2 x}}{2}$ and $x$ .

$- \frac{e^{- 2 x} x}{2} + \frac{- 3 e^{- 2 x}}{2} - \frac{1}{4} e^{u} + C$

Step 13.2.2

Move the negative in front of the fraction.

$- \frac{e^{- 2 x} x}{2} - \frac{3 e^{- 2 x}}{2} - \frac{1}{4} e^{u} + C$

Step 14

Replace all occurrences of $u$ with $- 2 x$ .

$- \frac{e^{- 2 x} x}{2} - \frac{3 e^{- 2 x}}{2} - \frac{1}{4} e^{- 2 x} + C$

Step 15

Combine $e^{- 2 x}$ and $\frac{1}{4}$ .

$- \frac{e^{- 2 x} x}{2} - \frac{3 e^{- 2 x}}{2} - \frac{e^{- 2 x}}{4} + C$

Step 16

Reorder terms.

$- \frac{1}{2} e^{- 2 x} x - \frac{3}{2} e^{- 2 x} - \frac{1}{4} e^{- 2 x} + C$

Step 17

The answer is the antiderivative of the function $f (x) = (x + 3) e^{- 2 x}$ .

$F (x) =$ $- \frac{1}{2} e^{- 2 x} x - \frac{3}{2} e^{- 2 x} - \frac{1}{4} e^{- 2 x} + C$