Calculus Examples

$\int 18 \sqrt{y} d y$

Step 1

Since $18$ is constant with respect to $y$ , move $18$ out of the integral.

$18 \int \sqrt{y} d y$

Step 2

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

$18 \int y^{\frac{1}{2}} d y$

Step 3

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

$18 (\frac{2}{3} y^{\frac{3}{2}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $18 (\frac{2}{3} y^{\frac{3}{2}} + C)$ as $18 (\frac{2}{3}) y^{\frac{3}{2}} + C$ .

$18 (\frac{2}{3}) y^{\frac{3}{2}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $18$ and $\frac{2}{3}$ .

$\frac{18 \cdot 2}{3} y^{\frac{3}{2}} + C$

Step 4.2.2

Multiply $18$ by $2$ .

$\frac{36}{3} y^{\frac{3}{2}} + C$

Step 4.2.3

Cancel the common factor of $36$ and $3$ .

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Step 4.2.3.1

Factor $3$ out of $36$ .

$\frac{3 \cdot 12}{3} y^{\frac{3}{2}} + C$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 12}{3 (1)} y^{\frac{3}{2}} + C$

Step 4.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 12}{3 \cdot 1} y^{\frac{3}{2}} + C$

Step 4.2.3.2.3

Rewrite the expression.

$\frac{12}{1} y^{\frac{3}{2}} + C$

Step 4.2.3.2.4

Divide $12$ by $1$ .

$12 y^{\frac{3}{2}} + C$