Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Move the negative in front of the fraction.

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

Step 7

Expand $(x - 1) x^{- \frac{1}{3}}$ .

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Step 7.1

Apply the distributive property.

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

Step 7.2

Raise $x$ to the power of $1$ .

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

Step 7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.4

Write $1$ as a fraction with a common denominator.

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

Step 7.5

Combine the numerators over the common denominator.

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

Step 7.6

Subtract $1$ from $3$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

$\frac{3}{5} x^{\frac{5}{3}} + C + \int - 1 x^{- \frac{1}{3}} d x$

Step 10

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

$\frac{3}{5} x^{\frac{5}{3}} + C - \int x^{- \frac{1}{3}} d x$

Step 11

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

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

Step 12

Simplify.

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

Step 13

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

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