Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

Let $u = x^{\frac{1}{3}} - 3$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{\frac{1}{3}} - 3$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Combine the numerators over the common denominator.

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

Step 2.1.3.5

Simplify the numerator.

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Step 2.1.3.5.1

Multiply $- 1$ by $3$ .

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

Step 2.1.3.5.2

Subtract $3$ from $1$ .

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

Step 2.1.3.6

Move the negative in front of the fraction.

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

Step 2.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 2.1.5

Simplify.

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Step 2.1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}} + 0$

Step 2.1.5.2

Combine terms.

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Step 2.1.5.2.1

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

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

Step 2.1.5.2.2

Add $\frac{1}{3 x^{\frac{2}{3}}}$ and $0$ .

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

Step 2.2

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

$\int 3 u d u$

Step 3

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

$3 \int u d u$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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