Calculus Examples

$\int \frac{5 \sqrt{x}}{2} d x$

Step 1

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

$\frac{5}{2} \int \sqrt{x} d x$

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{5}{2}$ by $\frac{2}{3}$ .

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

Step 4.2.2

Multiply $5$ by $2$ .

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

Step 4.2.3

Multiply $2$ by $3$ .

$\frac{10}{6} x^{\frac{3}{2}} + C$

Step 4.2.4

Cancel the common factor of $10$ and $6$ .

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Step 4.2.4.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{6} x^{\frac{3}{2}} + C$

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $2$ out of $6$ .

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

Step 4.2.4.2.2

Cancel the common factor.

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

Step 4.2.4.2.3

Rewrite the expression.

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