Calculus Examples

$\int x^{2} e^{- 3 x} d x$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

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

Step 2.2

Combine $x^{2}$ and $\frac{e^{- 3 x}}{3}$ .

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

Step 3

Since $- \frac{1}{3} \cdot 2$ is constant with respect to $x$ , move $- \frac{1}{3} \cdot 2$ out of the integral.

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

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 4.3

Move the negative in front of the fraction.

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

Step 4.4

Multiply $- 1$ by $- 1$ .

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

Step 4.5

Multiply $\frac{2}{3}$ by $1$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

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

Step 6.2

Combine $x$ and $\frac{e^{- 3 x}}{3}$ .

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

Step 6.3

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Multiply $\int \frac{e^{- 3 x}}{3} d x$ by $1$ .

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

Step 9

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

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

Step 10

Let $u = - 3 x$ . Then $d u = - 3 d x$ , so $- \frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 10.1

Let $u = - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 10.1.1

Differentiate $- 3 x$ .

$\frac{d}{d x} [- 3 x]$

Step 10.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 10.1.3

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

$- 3 \cdot 1$

Step 10.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 10.2

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

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

Step 11

Simplify.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

Combine $e^{u}$ and $\frac{1}{3}$ .

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

Step 12

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

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

Step 13

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 14

Simplify.

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Step 14.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 14.2

Multiply $3$ by $3$ .

$- \frac{x^{2} e^{- 3 x}}{3} + \frac{2}{3} (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} \int e^{u} d u)$

Step 15

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

$- \frac{x^{2} e^{- 3 x}}{3} + \frac{2}{3} (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u} + C))$

Step 16

Simplify.

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Step 16.1

Rewrite $- \frac{x^{2} e^{- 3 x}}{3} + \frac{2}{3} (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u} + C))$ as $- \frac{1}{3} x^{2} e^{- 3 x} + \frac{2}{3} (- \frac{1}{3} x e^{- 3 x} - \frac{1}{9} e^{u}) + C$ .

$- \frac{1}{3} x^{2} e^{- 3 x} + \frac{2}{3} (- \frac{1}{3} x e^{- 3 x} - \frac{1}{9} e^{u}) + C$

Step 16.2

Simplify.

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Step 16.2.1

Combine $x^{2}$ and $\frac{1}{3}$ .

$- \frac{x^{2}}{3} e^{- 3 x} + \frac{2}{3} (- \frac{1}{3} x e^{- 3 x} - \frac{1}{9} e^{u}) + C$

Step 16.2.2

Combine $e^{- 3 x}$ and $\frac{x^{2}}{3}$ .

$- \frac{e^{- 3 x} x^{2}}{3} + \frac{2}{3} (- \frac{1}{3} x e^{- 3 x} - \frac{1}{9} e^{u}) + C$

Step 16.2.3

Combine $x$ and $\frac{1}{3}$ .

$- \frac{e^{- 3 x} x^{2}}{3} + \frac{2}{3} (- \frac{x}{3} e^{- 3 x} - \frac{1}{9} e^{u}) + C$

Step 16.2.4

Combine $e^{- 3 x}$ and $\frac{x}{3}$ .

$- \frac{e^{- 3 x} x^{2}}{3} + \frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{1}{9} e^{u}) + C$

Step 16.2.5

Combine $e^{u}$ and $\frac{1}{9}$ .

$- \frac{e^{- 3 x} x^{2}}{3} + \frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9}) + C$

Step 16.2.6

To write $\frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{e^{- 3 x} x^{2}}{3} + \frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9}) \cdot \frac{3}{3} + C$

Step 16.2.7

Combine $\frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})$ and $\frac{3}{3}$ .

$- \frac{e^{- 3 x} x^{2}}{3} + \frac{\frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9}) \cdot 3}{3} + C$

Step 16.2.8

Combine the numerators over the common denominator.

$\frac{- e^{- 3 x} x^{2} + \frac{2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9}) \cdot 3}{3} + C$

Step 16.2.9

Combine $3$ and $\frac{2}{3}$ .

$\frac{- e^{- 3 x} x^{2} + \frac{3 \cdot 2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 16.2.10

Multiply $3$ by $2$ .

$\frac{- e^{- 3 x} x^{2} + \frac{6}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 16.2.11

Cancel the common factor of $6$ and $3$ .

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Step 16.2.11.1

Factor $3$ out of $6$ .

$\frac{- e^{- 3 x} x^{2} + \frac{3 \cdot 2}{3} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 16.2.11.2

Cancel the common factors.

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Step 16.2.11.2.1

Factor $3$ out of $3$ .

$\frac{- e^{- 3 x} x^{2} + \frac{3 \cdot 2}{3 (1)} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 16.2.11.2.2

Cancel the common factor.

$\frac{- e^{- 3 x} x^{2} + \frac{3 \cdot 2}{3 \cdot 1} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 16.2.11.2.3

Rewrite the expression.

$\frac{- e^{- 3 x} x^{2} + \frac{2}{1} (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 16.2.11.2.4

Divide $2$ by $1$ .

$\frac{- e^{- 3 x} x^{2} + 2 (- \frac{e^{- 3 x} x}{3} - \frac{e^{u}}{9})}{3} + C$

Step 17

Replace all occurrences of $u$ with $- 3 x$ .

$\frac{- e^{- 3 x} x^{2} + 2 (- \frac{e^{- 3 x} x}{3} - \frac{e^{- 3 x}}{9})}{3} + C$

Step 18

Reorder terms.

$\frac{1}{3} (- e^{- 3 x} x^{2} + 2 (- \frac{1}{3} e^{- 3 x} x - \frac{1}{9} e^{- 3 x})) + C$