Calculus Examples

$\int x^{2} e^{11 x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{11 x}$ .

$x^{2} (\frac{1}{11} e^{11 x}) - \int \frac{1}{11} e^{11 x} (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{11}$ and $e^{11 x}$ .

$x^{2} \frac{e^{11 x}}{11} - \int \frac{1}{11} e^{11 x} (2 x) d x$

Step 2.2

Combine $x^{2}$ and $\frac{e^{11 x}}{11}$ .

$\frac{x^{2} e^{11 x}}{11} - \int \frac{1}{11} e^{11 x} (2 x) d x$

Step 3

Since $\frac{1}{11} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{11} \cdot 2$ out of the integral.

$\frac{x^{2} e^{11 x}}{11} - (\frac{1}{11} \cdot 2 \int e^{11 x} (x) d x)$

Step 4

Combine $\frac{1}{11}$ and $2$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} \int e^{11 x} (x) d x$

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{11 x}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (x (\frac{1}{11} e^{11 x}) - \int \frac{1}{11} e^{11 x} d x)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{11}$ and $e^{11 x}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (x \frac{e^{11 x}}{11} - \int \frac{1}{11} e^{11 x} d x)$

Step 6.2

Combine $x$ and $\frac{e^{11 x}}{11}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \int \frac{1}{11} e^{11 x} d x)$

Step 6.3

Combine $\frac{1}{11}$ and $e^{11 x}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \int \frac{e^{11 x}}{11} d x)$

Step 7

Since $\frac{1}{11}$ is constant with respect to $x$ , move $\frac{1}{11}$ out of the integral.

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - (\frac{1}{11} \int e^{11 x} d x))$

Step 8

Let $u = 11 x$ . Then $d u = 11 d x$ , so $\frac{1}{11} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 11 x$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $11 x$ .

$\frac{d}{d x} [11 x]$

Step 8.1.2

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$11 \frac{d}{d x} [x]$

Step 8.1.3

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

$11 \cdot 1$

Step 8.1.4

Multiply $11$ by $1$ .

$11$

Step 8.2

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

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{11} \int e^{u} \frac{1}{11} d u)$

Step 9

Combine $e^{u}$ and $\frac{1}{11}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{11} \int \frac{e^{u}}{11} d u)$

Step 10

Since $\frac{1}{11}$ is constant with respect to $u$ , move $\frac{1}{11}$ out of the integral.

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{11} (\frac{1}{11} \int e^{u} d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{11}$ by $\frac{1}{11}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{11 \cdot 11} \int e^{u} d u)$

Step 11.2

Multiply $11$ by $11$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{121} \int e^{u} d u)$

Step 12

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

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{121} (e^{u} + C))$

Step 13

Simplify.

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Step 13.1

Rewrite $\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{121} (e^{u} + C))$ as $\frac{1}{11} x^{2} e^{11 x} - \frac{2}{11} (\frac{1}{11} x e^{11 x} - \frac{1}{121} e^{u}) + C$ .

$\frac{1}{11} x^{2} e^{11 x} - \frac{2}{11} (\frac{1}{11} x e^{11 x} - \frac{1}{121} e^{u}) + C$

Step 13.2

Simplify.

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Step 13.2.1

Combine $\frac{1}{11}$ and $x^{2}$ .

$\frac{x^{2}}{11} e^{11 x} - \frac{2}{11} (\frac{1}{11} x e^{11 x} - \frac{1}{121} e^{u}) + C$

Step 13.2.2

Combine $\frac{x^{2}}{11}$ and $e^{11 x}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{1}{11} x e^{11 x} - \frac{1}{121} e^{u}) + C$

Step 13.2.3

Combine $\frac{1}{11}$ and $x$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x}{11} e^{11 x} - \frac{1}{121} e^{u}) + C$

Step 13.2.4

Combine $\frac{x}{11}$ and $e^{11 x}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{1}{121} e^{u}) + C$

Step 13.2.5

Combine $e^{u}$ and $\frac{1}{121}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121}) + C$

Step 13.2.6

To write $- \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$\frac{x^{2} e^{11 x}}{11} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121}) \cdot \frac{11}{11} + C$

Step 13.2.7

Combine $- \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})$ and $\frac{11}{11}$ .

$\frac{x^{2} e^{11 x}}{11} + \frac{- \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121}) \cdot 11}{11} + C$

Step 13.2.8

Combine the numerators over the common denominator.

$\frac{x^{2} e^{11 x} - \frac{2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121}) \cdot 11}{11} + C$

Step 13.2.9

Multiply $11$ by $- 1$ .

$\frac{x^{2} e^{11 x} - 11 (\frac{2}{11}) (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.10

Combine $- 11$ and $\frac{2}{11}$ .

$\frac{x^{2} e^{11 x} + \frac{- 11 \cdot 2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.11

Multiply $- 11$ by $2$ .

$\frac{x^{2} e^{11 x} + \frac{- 22}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.12

Cancel the common factor of $- 22$ and $11$ .

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Step 13.2.12.1

Factor $11$ out of $- 22$ .

$\frac{x^{2} e^{11 x} + \frac{11 \cdot - 2}{11} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.12.2

Cancel the common factors.

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Step 13.2.12.2.1

Factor $11$ out of $11$ .

$\frac{x^{2} e^{11 x} + \frac{11 \cdot - 2}{11 (1)} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.12.2.2

Cancel the common factor.

$\frac{x^{2} e^{11 x} + \frac{11 \cdot - 2}{11 \cdot 1} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.12.2.3

Rewrite the expression.

$\frac{x^{2} e^{11 x} + \frac{- 2}{1} (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 13.2.12.2.4

Divide $- 2$ by $1$ .

$\frac{x^{2} e^{11 x} - 2 (\frac{x e^{11 x}}{11} - \frac{e^{u}}{121})}{11} + C$

Step 14

Replace all occurrences of $u$ with $11 x$ .

$\frac{x^{2} e^{11 x} - 2 (\frac{x e^{11 x}}{11} - \frac{e^{11 x}}{121})}{11} + C$

Step 15

Reorder terms.

$\frac{1}{11} (x^{2} e^{11 x} - 2 (\frac{1}{11} x e^{11 x} - \frac{1}{121} e^{11 x})) + C$