Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

Expand $(x^{2} - 1) x^{- \frac{1}{3}}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $3$ .

$\int x^{\frac{6 - 1}{3}} - 1 x^{- \frac{1}{3}} d x$

Step 4.6.2

Subtract $1$ from $6$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

By the Power Rule, the integral of $x^{\frac{5}{3}}$ with respect to $x$ is $\frac{3}{8} x^{\frac{8}{3}}$ .

$\frac{3}{8} x^{\frac{8}{3}} + C + \int - 1 x^{- \frac{1}{3}} d x$

Step 7

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

$\frac{3}{8} x^{\frac{8}{3}} + C - \int x^{- \frac{1}{3}} d x$

Step 8

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

$\frac{3}{8} x^{\frac{8}{3}} + C - (\frac{3}{2} x^{\frac{2}{3}} + C)$

Step 9

Simplify.

$\frac{3}{8} x^{\frac{8}{3}} - \frac{3}{2} x^{\frac{2}{3}} + C$