Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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

Step 8

Apply basic rules of exponents.

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Step 8.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 8.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 8.2.1

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

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

Step 8.2.2

Multiply $2$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{2} \cdot \frac{3}{2} x^{\frac{2}{3}} - - x^{- 1} + C$

Step 10.2

Simplify.

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Step 10.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$\frac{3}{2 \cdot 2} x^{\frac{2}{3}} - - x^{- 1} + C$

Step 10.2.2

Multiply $2$ by $2$ .

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

Step 10.2.3

Multiply $- 1$ by $- 1$ .

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

Step 10.2.4

Multiply $x^{- 1}$ by $1$ .

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

Step 11

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

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