Calculus Examples

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

Step 1

Write $(x^{\frac{1}{3}} - x^{\frac{2}{3}})$ as a function.

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

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.

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

Step 4

Remove parentheses.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

$\frac{3}{4} x^{\frac{4}{3}} + C + \int - x^{\frac{2}{3}} d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{3}{4} x^{\frac{4}{3}} + C - \int x^{\frac{2}{3}} d x$

Step 8

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

$\frac{3}{4} x^{\frac{4}{3}} + C - (\frac{3}{5} x^{\frac{5}{3}} + C)$

Step 9

Simplify.

$\frac{3}{4} x^{\frac{4}{3}} - \frac{3}{5} x^{\frac{5}{3}} + C$

Step 10

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

$F (x) =$ $\frac{3}{4} x^{\frac{4}{3}} - \frac{3}{5} x^{\frac{5}{3}} + C$