Calculus Examples

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

Step 1

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 2

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

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

Step 3

Let $u_{2} = \frac{u_{1}}{2}$ . Then $d u_{2} = \frac{1}{2} d u_{1}$ , so $2 d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.1

Let $u_{2} = \frac{u_{1}}{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 3.1.1

Differentiate $\frac{u_{1}}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$

Step 3.1.2

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{2}$ with respect to $u_{1}$ is $\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$

Step 3.1.3

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

$\frac{1}{2} \cdot 1$

Step 3.1.4

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

$\frac{1}{2}$

Step 3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int {u_{2}}^{2} e^{2 u_{2}} d u_{2}$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Move $2$ to the left of $e^{u_{3}}$ .

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

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Divide $e^{u_{3}}$ by $1$ .

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

$\frac{{u_{2}}^{2} e^{u_{3}}}{2} - e^{u_{3}} \int u_{2} d u_{2}$

Step 8

By the Power Rule, the integral of $u_{2}$ with respect to $u_{2}$ is $\frac{1}{2} {u_{2}}^{2}$ .

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Rewrite $\frac{{u_{2}}^{2} e^{u_{3}}}{2} - e^{u_{3}} (\frac{{u_{2}}^{2}}{2} + C)$ as $\frac{{u_{2}}^{2} e^{u_{3}}}{2} - \frac{{u_{2}}^{2} e^{u_{3}}}{2} + C$ .

$\frac{{u_{2}}^{2} e^{u_{3}}}{2} - \frac{{u_{2}}^{2} e^{u_{3}}}{2} + C$

Step 10

Remove parentheses.

$\frac{1}{2} {u_{2}}^{2} e^{u_{3}} - \frac{1}{2} {u_{2}}^{2} e^{u_{3}} + C$

Step 11

Simplify.

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Step 11.1

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

$\frac{{u_{2}}^{2}}{2} e^{u_{3}} - \frac{1}{2} {u_{2}}^{2} e^{u_{3}} + C$

Step 11.2

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

$\frac{{u_{2}}^{2} e^{u_{3}}}{2} - \frac{1}{2} {u_{2}}^{2} e^{u_{3}} + C$

Step 11.3

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

$\frac{{u_{2}}^{2} e^{u_{3}}}{2} - \frac{{u_{2}}^{2}}{2} e^{u_{3}} + C$

Step 11.4

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

$\frac{{u_{2}}^{2} e^{u_{3}}}{2} - \frac{e^{u_{3}} {u_{2}}^{2}}{2} + C$

Step 11.5

Subtract $\frac{e^{u_{3}} {u_{2}}^{2}}{2}$ from $\frac{{u_{2}}^{2} e^{u_{3}}}{2}$ .

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Step 11.5.1

Reorder $e^{u_{3}}$ and ${u_{2}}^{2}$ .

$\frac{{u_{2}}^{2} e^{u_{3}}}{2} - \frac{{u_{2}}^{2} e^{u_{3}}}{2} + C$

Step 11.5.2

Subtract $\frac{{u_{2}}^{2} e^{u_{3}}}{2}$ from $\frac{{u_{2}}^{2} e^{u_{3}}}{2}$ .

$0 + C$

Step 11.6

Add $0$ and $C$ .

$C$