Calculus Examples

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

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Combine $5$ and $\frac{4}{5}$ .

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

Step 4.2.2

Multiply $5$ by $4$ .

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

Step 4.2.3

Cancel the common factor of $20$ and $5$ .

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Step 4.2.3.1

Factor $5$ out of $20$ .

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

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $5$ out of $5$ .

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

Step 4.2.3.2.2

Cancel the common factor.

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

Step 4.2.3.2.3

Rewrite the expression.

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

Step 4.2.3.2.4

Divide $4$ by $1$ .

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