Calculus Examples

$\int {(\frac{1}{3} y)}^{\frac{1}{2}} d y$

Step 1

Let $u = \frac{1}{3} y$ . Then $d u = \frac{1}{3} d y$ , so $3 d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{1}{3} y$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $\frac{1}{3} y$ .

$\frac{d}{d y} [\frac{1}{3} y]$

Step 1.1.2

Since $\frac{1}{3}$ is constant with respect to $y$ , the derivative of $\frac{1}{3} y$ with respect to $y$ is $\frac{1}{3} \frac{d}{d y} [y]$ .

$\frac{1}{3} \frac{d}{d y} [y]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\frac{1}{3} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$\int u^{\frac{1}{2}} (1 \cdot 3) d u$

Step 2.2

Multiply $3$ by $1$ .

$\int u^{\frac{1}{2}} \cdot 3 d u$

Step 2.3

Move $3$ to the left of $u^{\frac{1}{2}}$ .

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

Step 3

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{3 \cdot 2}{3} u^{\frac{3}{2}} + C$

Step 5.2.2

Multiply $3$ by $2$ .

$\frac{6}{3} u^{\frac{3}{2}} + C$

Step 5.2.3

Cancel the common factor of $6$ and $3$ .

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Step 5.2.3.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 2}{3} u^{\frac{3}{2}} + C$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $3$ out of $3$ .

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

Step 5.2.3.2.2

Cancel the common factor.

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

Step 5.2.3.2.3

Rewrite the expression.

$\frac{2}{1} u^{\frac{3}{2}} + C$

Step 5.2.3.2.4

Divide $2$ by $1$ .

$2 u^{\frac{3}{2}} + C$

Step 6

Replace all occurrences of $u$ with $\frac{1}{3} y$ .

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