Calculus Examples

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

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.

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

Step 3

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- 3 \int \sqrt[3]{x} d x$

Step 4

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $- 3 (\frac{3}{4} x^{\frac{4}{3}} + C)$ as $- 3 (\frac{3}{4}) x^{\frac{4}{3}} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $- 3$ and $\frac{3}{4}$ .

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

Step 6.2.2

Multiply $- 3$ by $3$ .

$\frac{- 9}{4} x^{\frac{4}{3}} + C$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{9}{4} x^{\frac{4}{3}} + C$

Step 7

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

$F (x) =$ $- \frac{9}{4} x^{\frac{4}{3}} + C$