Calculus Examples

$\frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}$

Step 1

Write $\frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}$ as a function.

$f (x) = \frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}$

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.

$F (x) = \int \frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{4}{\sqrt[3]{x}} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \frac{1}{\sqrt[3]{x}} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 6

Apply basic rules of exponents.

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Step 6.1

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

$4 \int \frac{1}{x^{\frac{1}{3}}} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 6.2

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

$4 \int {(x^{\frac{1}{3}})}^{- 1} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 6.3

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

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Step 6.3.1

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

$4 \int x^{\frac{1}{3} \cdot - 1} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 6.3.2

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

$4 \int x^{\frac{- 1}{3}} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 6.3.3

Move the negative in front of the fraction.

$4 \int x^{- \frac{1}{3}} d x + \int - 4 \sqrt[3]{x^{2}} d x$

Step 7

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

$4 (\frac{3}{2} x^{\frac{2}{3}} + C) + \int - 4 \sqrt[3]{x^{2}} d x$

Step 8

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$4 (\frac{3}{2} x^{\frac{2}{3}} + C) - 4 \int \sqrt[3]{x^{2}} d x$

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

Combine $4$ and $\frac{3}{2}$ .

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

Step 11.2.2

Multiply $4$ by $3$ .

$\frac{12}{2} x^{\frac{2}{3}} - 4 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 11.2.3

Cancel the common factor of $12$ and $2$ .

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Step 11.2.3.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 6}{2} x^{\frac{2}{3}} - 4 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 11.2.3.2

Cancel the common factors.

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Step 11.2.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 6}{2 (1)} x^{\frac{2}{3}} - 4 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 11.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 6}{2 \cdot 1} x^{\frac{2}{3}} - 4 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 11.2.3.2.3

Rewrite the expression.

$\frac{6}{1} x^{\frac{2}{3}} - 4 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 11.2.3.2.4

Divide $6$ by $1$ .

$6 x^{\frac{2}{3}} - 4 (\frac{3}{5}) x^{\frac{5}{3}} + C$

Step 11.2.4

Combine $- 4$ and $\frac{3}{5}$ .

$6 x^{\frac{2}{3}} + \frac{- 4 \cdot 3}{5} x^{\frac{5}{3}} + C$

Step 11.2.5

Multiply $- 4$ by $3$ .

$6 x^{\frac{2}{3}} + \frac{- 12}{5} x^{\frac{5}{3}} + C$

Step 11.2.6

Move the negative in front of the fraction.

$6 x^{\frac{2}{3}} - \frac{12}{5} x^{\frac{5}{3}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = \frac{4}{\sqrt[3]{x}} - 4 \sqrt[3]{x^{2}}$ .

$F (x) =$ $6 x^{\frac{2}{3}} - \frac{12}{5} x^{\frac{5}{3}} + C$