Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

Apply the product rule to $\frac{1 - \sqrt[3]{x}}{x^{2}}$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

Multiply $2$ by $- 1$ .

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

Let $u = 1 - \sqrt[3]{x}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $1 - \sqrt[3]{x}$ .

$\frac{d}{d x} [1 - \sqrt[3]{x}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$\frac{d}{d x} [1] + \frac{d}{d x} [- \sqrt[3]{x}]$

Step 2.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \sqrt[3]{x}]$

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

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

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

Step 2.1.3.6

Combine the numerators over the common denominator.

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

Step 2.1.3.7

Simplify the numerator.

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Step 2.1.3.7.1

Multiply $- 1$ by $3$ .

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

Step 2.1.3.7.2

Subtract $3$ from $1$ .

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

Step 2.1.3.8

Move the negative in front of the fraction.

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

Step 2.1.3.9

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

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

Step 2.1.3.10

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

$0 - \frac{1}{3 x^{\frac{2}{3}}}$

Step 2.1.4

Subtract $\frac{1}{3 x^{\frac{2}{3}}}$ from $0$ .

$- \frac{1}{3 x^{\frac{2}{3}}}$

Step 2.2

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

$\int - 1 \cdot 3 u^{\frac{1}{3}} d u$

Step 3

Multiply $- 1$ by $3$ .

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

Step 4

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

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

Step 5

By the Power Rule, the integral of $u^{\frac{1}{3}}$ with respect to $u$ is $\frac{3}{4} u^{\frac{4}{3}}$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $- 3$ and $\frac{3}{4}$ .

$\frac{- 3 \cdot 3}{4} u^{\frac{4}{3}} + C$

Step 6.2.2

Multiply $- 3$ by $3$ .

$\frac{- 9}{4} u^{\frac{4}{3}} + C$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{9}{4} u^{\frac{4}{3}} + C$

Step 7

Replace all occurrences of $u$ with $1 - \sqrt[3]{x}$ .

$- \frac{9}{4} {(1 - \sqrt[3]{x})}^{\frac{4}{3}} + C$