Calculus Examples

$\int (5 \sqrt{x}) d x$

Step 1

Remove parentheses.

$\int 5 \sqrt{x} d x$

Step 2

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

$5 \int \sqrt{x} d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4

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

$5 (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $5 (\frac{2}{3} x^{\frac{3}{2}} + C)$ as $5 (\frac{2}{3}) x^{\frac{3}{2}} + C$ .

$5 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 5.2

Simplify.

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Step 5.2.1

Combine $5$ and $\frac{2}{3}$ .

$\frac{5 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 5.2.2

Multiply $5$ by $2$ .

$\frac{10}{3} x^{\frac{3}{2}} + C$