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

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

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

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

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

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

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

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

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

Step 7

Since $\frac{1}{3}$ 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 8

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 8.1

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

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Step 8.1.1

Differentiate $3 x$ .

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

Step 8.1.2

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 8.1.3

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 8.1.4

Multiply $3$ by $1$ .

$3$

Step 8.2

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 9

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 10

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 11

Simplify.

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Step 11.1

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 11.2

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 12

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 13

Simplify.

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Step 13.1

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 13.2

Simplify.

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Step 13.2.1

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

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

Step 13.2.2

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

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

Step 13.2.3

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

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

Step 13.2.4

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

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

Step 13.2.5

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

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

Step 13.2.6

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

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

Step 13.2.7

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

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

Step 13.2.8

Combine the numerators over the common denominator.

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

Step 13.2.9

Multiply $3$ by $- 1$ .

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

Step 13.2.10

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

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

Step 13.2.11

Multiply $- 3$ by $2$ .

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

Step 13.2.12

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

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Step 13.2.12.1

Factor $3$ out of $- 6$ .

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

Step 13.2.12.2

Cancel the common factors.

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Step 13.2.12.2.1

Factor $3$ out of $3$ .

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

Step 13.2.12.2.2

Cancel the common factor.

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

Step 13.2.12.2.3

Rewrite the expression.

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

Step 13.2.12.2.4

Divide $- 2$ by $1$ .

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

Step 14

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

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

Step 15

Reorder terms.

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