Calculus Examples

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

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

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

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

Step 6

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

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

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 $d u$ .

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

Step 8

Simplify.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Multiply $2$ by $2$ .

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Simplify.

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Step 13.2.1

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

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

Step 13.2.2

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

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

Step 13.2.3

Combine $e^{u}$ and $\frac{1}{4}$ .

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

Apply the distributive property.

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

Step 15.2

Cancel the common factor of $2$ .

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Step 15.2.1

Move the leading negative in $- \frac{e^{- 2 x} x}{2}$ into the numerator.

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

Step 15.2.2

Cancel the common factor.

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

Step 15.2.3

Rewrite the expression.

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

Step 15.3

Cancel the common factor of $2$ .

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Step 15.3.1

Move the leading negative in $- \frac{e^{- 2 x}}{4}$ into the numerator.

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

Step 15.3.2

Factor $2$ out of $4$ .

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

Step 15.3.3

Cancel the common factor.

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

Step 15.3.4

Rewrite the expression.

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

Step 15.4

Move the negative in front of the fraction.

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

Step 15.5

Combine $- e^{- 2 x} x$ and $- \frac{e^{- 2 x}}{2}$ using a common denominator.

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Step 15.5.1

Move $e^{- 2 x}$ .

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

Step 15.5.2

To write $- x e^{- 2 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 15.5.3

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

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

Step 15.5.4

Combine the numerators over the common denominator.

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

Step 15.6

Simplify the numerator.

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Step 15.6.1

Factor $e^{- 2 x}$ out of $- x e^{- 2 x} \cdot 2 - e^{- 2 x}$ .

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Step 15.6.1.1

Factor $e^{- 2 x}$ out of $- x e^{- 2 x} \cdot 2$ .

$\frac{e^{- 2 x} (- x \cdot 2) - e^{- 2 x}}{2} + C$

Step 15.6.1.2

Factor $e^{- 2 x}$ out of $- e^{- 2 x}$ .

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

Step 15.6.1.3

Factor $e^{- 2 x}$ out of $e^{- 2 x} (- x \cdot 2) + e^{- 2 x} \cdot - 1$ .

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

Step 15.6.2

Multiply $2$ by $- 1$ .

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

Step 15.7

Factor $- 1$ out of $- 2 x$ .

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

Step 15.8

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 15.9

Factor $- 1$ out of $- (2 x) - 1 (1)$ .

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

Step 15.10

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

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

Step 15.11

Move the negative in front of the fraction.

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

Step 16

Remove parentheses.

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