Calculus Examples

$\sqrt[3]{x^{2}}$

Step 1

Write $\sqrt[3]{x^{2}}$ as a function.

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

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.

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

Step 4

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

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

Step 5

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$\frac{3}{5} x^{\frac{5}{3}} + C$

Step 6

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

$F (x) =$ $\frac{3}{5} x^{\frac{5}{3}} + C$