Calculus Examples

$f (x) = 2 x^{\frac{2}{3}} - 5 x^{3}$

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.

$F (x) = \int 2 x^{\frac{2}{3}} - 5 x^{3} d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 x^{\frac{2}{3}} d x + \int - 5 x^{3} d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{\frac{2}{3}} d x + \int - 5 x^{3} d x$

Step 5

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

$2 (\frac{3}{5} x^{\frac{5}{3}} + C) + \int - 5 x^{3} d x$

Step 6

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$2 (\frac{3}{5} x^{\frac{5}{3}} + C) - 5 \int x^{3} d x$

Step 7

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

$2 (\frac{3}{5} x^{\frac{5}{3}} + C) - 5 (\frac{1}{4} x^{4} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$2 (\frac{3}{5}) x^{\frac{5}{3}} - 5 (\frac{1}{4}) x^{4} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{3}{5}$ .

$\frac{2 \cdot 3}{5} x^{\frac{5}{3}} - 5 (\frac{1}{4}) x^{4} + C$

Step 8.2.2

Multiply $2$ by $3$ .

$\frac{6}{5} x^{\frac{5}{3}} - 5 (\frac{1}{4}) x^{4} + C$

Step 8.2.3

Combine $- 5$ and $\frac{1}{4}$ .

$\frac{6}{5} x^{\frac{5}{3}} + \frac{- 5}{4} x^{4} + C$

Step 8.2.4

Move the negative in front of the fraction.

$\frac{6}{5} x^{\frac{5}{3}} - \frac{5}{4} x^{4} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 2 x^{\frac{2}{3}} - 5 x^{3}$ .

$F (x) =$ $\frac{6}{5} x^{\frac{5}{3}} - \frac{5}{4} x^{4} + C$