Calculus Examples

$\int \frac{\sqrt{x^{3}}}{2} d x$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sqrt{x^{3}} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$\frac{1}{2} \int x^{\frac{3}{2}} d x$

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{1}{2} (\frac{2}{5} x^{\frac{5}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{5} x^{\frac{5}{2}} + C$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{5}$ .

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

Step 4.2.2

Multiply $2$ by $5$ .

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

Step 4.2.3

Cancel the common factor of $2$ and $10$ .

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Step 4.2.3.1

Factor $2$ out of $2$ .

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

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $2$ out of $10$ .

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

Step 4.2.3.2.2

Cancel the common factor.

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

Step 4.2.3.2.3

Rewrite the expression.

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