Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

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

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

Step 4.3

Cancel the common factor of $- 2$ and $2$ .

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Step 4.3.1

Factor $2$ out of $- 2$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $2$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.3.2.4

Divide $- 1$ by $1$ .

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

Step 4.4

Multiply $- 1$ by $- 1$ .

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

Step 4.5

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

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

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

Step 9

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 9.1

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

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Step 9.1.1

Differentiate $- 2 x$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 9.2

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

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

Step 10

Simplify.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Multiply $2$ by $2$ .

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

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

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

Step 15.2

Simplify.

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Step 15.2.1

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

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

Step 15.2.2

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

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

Step 15.2.3

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

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

Step 15.2.4

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

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

Step 16

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

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

Step 17

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

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

Step 18

Reorder terms.

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

Step 19

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

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

Step 20

Remove parentheses.

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