Calculus Examples

$\int \frac{x}{e^{2 x}} d x$

Step 1

Simplify the expression.

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Step 1.1

Negate the exponent of $e^{2 x}$ and move it out of the denominator.

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

Step 1.2

Multiply the exponents in ${(e^{2 x})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x e^{2 x \cdot - 1} d x$

Step 1.2.2

Multiply $- 1$ by $2$ .

$\int x e^{- 2 x} d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

$- \frac{x e^{- 2 x}}{2} - \int - \frac{1}{2} e^{- 2 x} d x$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

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

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

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $- 2 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

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

Step 7.2

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

$- \frac{x e^{- 2 x}}{2} - \frac{1}{2 \cdot 2} \int e^{u} d u$

Step 10.2

Multiply $2$ by $2$ .

$- \frac{x e^{- 2 x}}{2} - \frac{1}{4} \int e^{u} d u$

Step 11

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

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Simplify.

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Step 12.2.1

Combine $x$ and $\frac{1}{2}$ .

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

Step 12.2.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

Reorder terms.

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