Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

Move $x^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.3

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

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Step 1.3.1

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

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

Step 1.3.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 1.3.2.1

Combine $\frac{2}{3}$ and $- 1$ .

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

Step 1.3.2.2

Multiply $2$ by $- 1$ .

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

Step 1.3.3

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify.

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Add $1$ and $3$ .

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