Calculus Examples

$\int \frac{5 \sqrt[4]{x}}{2} d x$

Step 1

Since $\frac{5}{2}$ is constant with respect to $x$ , move $\frac{5}{2}$ out of the integral.

$\frac{5}{2} \int \sqrt[4]{x} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{5}{2} \int x^{\frac{1}{4}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$\frac{5}{2} (\frac{4}{5} x^{\frac{5}{4}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{5}{2} (\frac{4}{5} x^{\frac{5}{4}} + C)$ as $\frac{5}{2} \cdot \frac{4}{5} x^{\frac{5}{4}} + C$ .

$\frac{5}{2} \cdot \frac{4}{5} x^{\frac{5}{4}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{5}{2}$ by $\frac{4}{5}$ .

$\frac{5 \cdot 4}{2 \cdot 5} x^{\frac{5}{4}} + C$

Step 4.2.2

Multiply $5$ by $4$ .

$\frac{20}{2 \cdot 5} x^{\frac{5}{4}} + C$

Step 4.2.3

Multiply $2$ by $5$ .

$\frac{20}{10} x^{\frac{5}{4}} + C$

Step 4.2.4

Cancel the common factor of $20$ and $10$ .

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Step 4.2.4.1

Factor $10$ out of $20$ .

$\frac{10 \cdot 2}{10} x^{\frac{5}{4}} + C$

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $10$ out of $10$ .

$\frac{10 \cdot 2}{10 (1)} x^{\frac{5}{4}} + C$

Step 4.2.4.2.2

Cancel the common factor.

$\frac{10 \cdot 2}{10 \cdot 1} x^{\frac{5}{4}} + C$

Step 4.2.4.2.3

Rewrite the expression.

$\frac{2}{1} x^{\frac{5}{4}} + C$

Step 4.2.4.2.4

Divide $2$ by $1$ .

$2 x^{\frac{5}{4}} + C$