Calculus Examples

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

Step 1

Write $\frac{1}{3 \sqrt[3]{x}}$ as a function.

$f (x) = \frac{1}{3 \sqrt[3]{x}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{1}{3 \sqrt[3]{x}} d x$

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $3$ by $2$ .

$\frac{3}{6} x^{\frac{2}{3}} + C$

Step 7.2.3

Cancel the common factor of $3$ and $6$ .

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Step 7.2.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{6} x^{\frac{2}{3}} + C$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $3$ out of $6$ .

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

Step 7.2.3.2.2

Cancel the common factor.

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

Step 7.2.3.2.3

Rewrite the expression.

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

Step 8

The answer is the antiderivative of the function $f (x) = \frac{1}{3 \sqrt[3]{x}}$ .

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