Calculus Examples

$\int (\frac{1}{\sqrt[5]{x^{3}}}) d x$

Step 1

Remove parentheses.

$\int \frac{1}{\sqrt[5]{x^{3}}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

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

Step 2.2

Move $x^{\frac{3}{5}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3

Multiply the exponents in ${(x^{\frac{3}{5}})}^{- 1}$ .

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Step 2.3.1

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

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

Step 2.3.2

Multiply $\frac{3}{5} \cdot - 1$ .

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Step 2.3.2.1

Combine $\frac{3}{5}$ and $- 1$ .

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

Step 2.3.2.2

Multiply $3$ by $- 1$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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