Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Cancel the common factor of $2$ .

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Step 6.6.1

Cancel the common factor.

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

Step 6.6.2

Divide $e^{2 x} x$ by $1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

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 10.1

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

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Step 10.1.1

Differentiate $2 x$ .

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

Step 10.1.2

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 10.1.3

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

$2 \cdot 1$

Step 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Multiply $2$ by $2$ .

$\frac{x^{3} e^{2 x}}{2} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u} d u))$

Step 14

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

$\frac{x^{3} e^{2 x}}{2} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C)))$

Step 15

Simplify.

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Step 15.1

Rewrite $\frac{x^{3} e^{2 x}}{2} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - (\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C)))$ as $\frac{x^{3} e^{2 x}}{2} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4}) + C$ .

$\frac{x^{3} e^{2 x}}{2} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4}) + C$

Step 15.2

Simplify.

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Step 15.2.1

To write $- \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{3} e^{2 x}}{2} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4}) \cdot \frac{2}{2} + C$

Step 15.2.2

Combine $- \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})$ and $\frac{2}{2}$ .

$\frac{x^{3} e^{2 x}}{2} + \frac{- \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4}) \cdot 2}{2} + C$

Step 15.2.3

Combine the numerators over the common denominator.

$\frac{x^{3} e^{2 x} - \frac{3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4}) \cdot 2}{2} + C$

Step 15.2.4

Multiply $2$ by $- 1$ .

$\frac{x^{3} e^{2 x} - 2 (\frac{3}{2}) (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.5

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

$\frac{x^{3} e^{2 x} + \frac{- 2 \cdot 3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.6

Multiply $- 2$ by $3$ .

$\frac{x^{3} e^{2 x} + \frac{- 6}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.7

Cancel the common factor of $- 6$ and $2$ .

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Step 15.2.7.1

Factor $2$ out of $- 6$ .

$\frac{x^{3} e^{2 x} + \frac{2 \cdot - 3}{2} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.7.2

Cancel the common factors.

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Step 15.2.7.2.1

Factor $2$ out of $2$ .

$\frac{x^{3} e^{2 x} + \frac{2 \cdot - 3}{2 (1)} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.7.2.2

Cancel the common factor.

$\frac{x^{3} e^{2 x} + \frac{2 \cdot - 3}{2 \cdot 1} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.7.2.3

Rewrite the expression.

$\frac{x^{3} e^{2 x} + \frac{- 3}{1} (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 15.2.7.2.4

Divide $- 3$ by $1$ .

$\frac{x^{3} e^{2 x} - 3 (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{u}}{4})}{2} + C$

Step 16

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

$\frac{x^{3} e^{2 x} - 3 (\frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{2 x}}{4})}{2} + C$

Step 17

Reorder terms.

$\frac{1}{2} (x^{3} e^{2 x} - 3 (\frac{1}{2} x^{2} e^{2 x} - \frac{1}{2} x e^{2 x} + \frac{1}{4} e^{2 x})) + C$