Calculus Examples

$\sqrt[3]{x} + \frac{1}{\sqrt[3]{x}}$

Step 1

Write $\sqrt[3]{x} + \frac{1}{\sqrt[3]{x}}$ as a function.

$f (x) = \sqrt[3]{x} + \frac{1}{\sqrt[3]{x}}$

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.

$F (x) = \int \sqrt[3]{x} + \frac{1}{\sqrt[3]{x}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \sqrt[3]{x} d x + \int \frac{1}{\sqrt[3]{x}} d x$

Step 5

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

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

Step 6

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

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

Step 7

Apply basic rules of exponents.

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Step 7.1

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

$\frac{3}{4} x^{\frac{4}{3}} + C + \int \frac{1}{x^{\frac{1}{3}}} d x$

Step 7.2

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

$\frac{3}{4} x^{\frac{4}{3}} + C + \int {(x^{\frac{1}{3}})}^{- 1} d x$

Step 7.3

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

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Step 7.3.1

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

$\frac{3}{4} x^{\frac{4}{3}} + C + \int x^{\frac{1}{3} \cdot - 1} d x$

Step 7.3.2

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

$\frac{3}{4} x^{\frac{4}{3}} + C + \int x^{\frac{- 1}{3}} d x$

Step 7.3.3

Move the negative in front of the fraction.

$\frac{3}{4} x^{\frac{4}{3}} + C + \int x^{- \frac{1}{3}} d x$

Step 8

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

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

Step 9

Simplify.

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

Step 10

The answer is the antiderivative of the function $f (x) = \sqrt[3]{x} + \frac{1}{\sqrt[3]{x}}$ .

$F (x) =$ $\frac{3}{4} x^{\frac{4}{3}} + \frac{3}{2} x^{\frac{2}{3}} + C$