Calculus Examples

$\int \frac{1}{6 \sqrt[4]{x}} d x$

Step 1

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int \frac{1}{\sqrt[4]{x}} d x$

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 $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{1}{6} \int \frac{1}{x^{\frac{1}{4}}} d x$

Step 2.2

Move $x^{\frac{1}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{6} \int {(x^{\frac{1}{4}})}^{- 1} d x$

Step 2.3

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 1}$ .

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Step 2.3.1

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

$\frac{1}{6} \int x^{\frac{1}{4} \cdot - 1} d x$

Step 2.3.2

Combine $\frac{1}{4}$ and $- 1$ .

$\frac{1}{6} \int x^{\frac{- 1}{4}} d x$

Step 2.3.3

Move the negative in front of the fraction.

$\frac{1}{6} \int x^{- \frac{1}{4}} d x$

Step 3

By the Power Rule, the integral of $x^{- \frac{1}{4}}$ with respect to $x$ is $\frac{4}{3} x^{\frac{3}{4}}$ .

$\frac{1}{6} (\frac{4}{3} x^{\frac{3}{4}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{1}{6} (\frac{4}{3} x^{\frac{3}{4}} + C)$ as $\frac{1}{6} \cdot \frac{4}{3} x^{\frac{3}{4}} + C$ .

$\frac{1}{6} \cdot \frac{4}{3} x^{\frac{3}{4}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{6}$ by $\frac{4}{3}$ .

$\frac{4}{6 \cdot 3} x^{\frac{3}{4}} + C$

Step 4.2.2

Multiply $6$ by $3$ .

$\frac{4}{18} x^{\frac{3}{4}} + C$

Step 4.2.3

Cancel the common factor of $4$ and $18$ .

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Step 4.2.3.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{18} x^{\frac{3}{4}} + C$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 2}{2 \cdot 9} x^{\frac{3}{4}} + C$

Step 4.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 9} x^{\frac{3}{4}} + C$

Step 4.2.3.2.3

Rewrite the expression.

$\frac{2}{9} x^{\frac{3}{4}} + C$