Calculus Examples

$\int \frac{\sqrt{2 + 9 \sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\int \frac{\sqrt{2 + 9 x^{\frac{1}{3}}}}{\sqrt[3]{x^{2}}} d x$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 + 9 x^{\frac{1}{3}}}$ as ${(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}}$ .

$\int \frac{{(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}}}{\sqrt[3]{x^{2}}} d x$

Step 1.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\int \frac{{(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}}}{x^{\frac{2}{3}}} d x$

Step 1.4

Move $x^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}} {(x^{\frac{2}{3}})}^{- 1} d x$

Step 1.5

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

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Step 1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int {(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}} x^{\frac{2}{3} \cdot - 1} d x$

Step 1.5.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 1.5.2.1

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

$\int {(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}} x^{\frac{2 \cdot - 1}{3}} d x$

Step 1.5.2.2

Multiply $2$ by $- 1$ .

$\int {(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}} x^{\frac{- 2}{3}} d x$

Step 1.5.3

Move the negative in front of the fraction.

$\int {(2 + 9 x^{\frac{1}{3}})}^{\frac{1}{2}} x^{- \frac{2}{3}} d x$

Step 2

Let $u = 2 + 9 x^{\frac{1}{3}}$ . Then $d u = \frac{3}{x^{\frac{2}{3}}} d x$ , so $- d u = \frac{- 3}{x^{\frac{2}{3}}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 + 9 x^{\frac{1}{3}}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 + 9 x^{\frac{1}{3}}$ .

$\frac{d}{d x} [2 + 9 x^{\frac{1}{3}}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $2 + 9 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [9 x^{\frac{1}{3}}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [9 x^{\frac{1}{3}}]$

Step 2.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [9 x^{\frac{1}{3}}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [9 x^{\frac{1}{3}}]$ .

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Step 2.1.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{1}{3}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$0 + 9 \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

$0 + 9 (\frac{1}{3} x^{\frac{1}{3} - 1})$

Step 2.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$0 + 9 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 2.1.3.4

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

$0 + 9 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.1.3.5

Combine the numerators over the common denominator.

$0 + 9 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Step 2.1.3.6

Simplify the numerator.

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Step 2.1.3.6.1

Multiply $- 1$ by $3$ .

$0 + 9 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Step 2.1.3.6.2

Subtract $3$ from $1$ .

$0 + 9 (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 2.1.3.7

Move the negative in front of the fraction.

$0 + 9 (\frac{1}{3} x^{- \frac{2}{3}})$

Step 2.1.3.8

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

$0 + 9 \frac{x^{- \frac{2}{3}}}{3}$

Step 2.1.3.9

Combine $9$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$0 + \frac{9 x^{- \frac{2}{3}}}{3}$

Step 2.1.3.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{9}{3 x^{\frac{2}{3}}}$

Step 2.1.3.11

Factor $3$ out of $9$ .

$0 + \frac{3 \cdot 3}{3 x^{\frac{2}{3}}}$

Step 2.1.3.12

Cancel the common factors.

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Step 2.1.3.12.1

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

$0 + \frac{3 \cdot 3}{3 (x^{\frac{2}{3}})}$

Step 2.1.3.12.2

Cancel the common factor.

$0 + \frac{3 \cdot 3}{3 x^{\frac{2}{3}}}$

Step 2.1.3.12.3

Rewrite the expression.

$0 + \frac{3}{x^{\frac{2}{3}}}$

Step 2.1.4

Add $0$ and $\frac{3}{x^{\frac{2}{3}}}$ .

$\frac{3}{x^{\frac{2}{3}}}$

Step 2.2

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

$\int \frac{1}{3} u^{\frac{1}{2}} d u$

Step 3

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

$\frac{1}{3} \int u^{\frac{1}{2}} d u$

Step 4

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{1}{3} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{3} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $\frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{3}$ by $\frac{2}{3}$ .

$\frac{2}{3 \cdot 3} u^{\frac{3}{2}} + C$

Step 5.2.2

Multiply $3$ by $3$ .

$\frac{2}{9} u^{\frac{3}{2}} + C$

Step 6

Replace all occurrences of $u$ with $2 + 9 x^{\frac{1}{3}}$ .

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