Calculus Examples

$\int e^{\sqrt{3 x + 9}} d x$

Step 1

Let $u_{1} = 3 x + 9$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 3 x + 9$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $3 x + 9$ .

$\frac{d}{d x} [3 x + 9]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x + 9$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [9]$

Step 1.1.3

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

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Step 1.1.3.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [9]$

Step 1.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [9]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [9]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$3 + 0$

Step 1.1.4.2

Add $3$ and $0$ .

$3$

Step 1.2

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

$\int e^{\sqrt{u_{1}}} \frac{1}{3} d u_{1}$

Step 2

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

$\int \frac{e^{\sqrt{u_{1}}}}{3} d u_{1}$

Step 3

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

$\frac{1}{3} \int e^{\sqrt{u_{1}}} d u_{1}$

Step 4

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

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

Step 5

Since $2$ is constant with respect to $u_{2}$ , move $2$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

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

Step 9

Simplify.

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

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

Replace all occurrences of $u_{2}$ with $\sqrt{u_{1}}$ .

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

Step 10.2

Replace all occurrences of $u_{1}$ with $3 x + 9$ .

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Factor $3$ out of $3 x + 9$ .

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Step 11.2.1

Factor $3$ out of $3 x$ .

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

Step 11.2.2

Factor $3$ out of $9$ .

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

Step 11.2.3

Factor $3$ out of $3 x + 3 \cdot 3$ .

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

Step 11.3

Factor $3$ out of $3 x + 9$ .

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Step 11.3.1

Factor $3$ out of $3 x$ .

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

Step 11.3.2

Factor $3$ out of $9$ .

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

Step 11.3.3

Factor $3$ out of $3 x + 3 \cdot 3$ .

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

Step 11.4

Simplify each term.

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Step 11.4.1

Factor $3$ out of $3 x + 9$ .

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Step 11.4.1.1

Factor $3$ out of $3 x$ .

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

Step 11.4.1.2

Factor $3$ out of $9$ .

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

Step 11.4.1.3

Factor $3$ out of $3 x + 3 \cdot 3$ .

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

Step 11.4.2

Multiply $\frac{2}{3} (\sqrt{3 (x + 3)} e^{\sqrt{3 (x + 3)}})$ .

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Step 11.4.2.1

Combine $\sqrt{3 (x + 3)}$ and $\frac{2}{3}$ .

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

Step 11.4.2.2

Combine $\frac{\sqrt{3 (x + 3)} \cdot 2}{3}$ and $e^{\sqrt{3 (x + 3)}}$ .

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

Step 11.4.3

Move $2$ to the left of $\sqrt{3 (x + 3)}$ .

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

Step 11.4.4

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

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

Step 11.5

Combine the numerators over the common denominator.

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

Step 11.6

Factor $2 e^{\sqrt{3 (x + 3)}}$ out of $2 \sqrt{3 (x + 3)} e^{\sqrt{3 (x + 3)}} - 2 e^{\sqrt{3 (x + 3)}}$ .

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Step 11.6.1

Factor $2 e^{\sqrt{3 (x + 3)}}$ out of $2 \sqrt{3 (x + 3)} e^{\sqrt{3 (x + 3)}}$ .

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

Step 11.6.2

Factor $2 e^{\sqrt{3 (x + 3)}}$ out of $- 2 e^{\sqrt{3 (x + 3)}}$ .

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

Step 11.6.3

Factor $2 e^{\sqrt{3 (x + 3)}}$ out of $2 e^{\sqrt{3 (x + 3)}} (\sqrt{3 (x + 3)}) + 2 e^{\sqrt{3 (x + 3)}} (- 1)$ .

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