Calculus Examples

$f (x) = \frac{1}{\sqrt[3]{x^{2}}}$

Step 1

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 2

Set up the integral to solve.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

Multiply $2$ by $- 1$ .

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

Step 3.3.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

The answer is the antiderivative of the function $f (x) = \frac{1}{\sqrt[3]{x^{2}}}$ .

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