Calculus Examples

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

Step 1

Let $u = 3 y^{2} - 2 y$ . Then $d u = (6 y - 2) d y$ , so $\frac{1}{2} d u = (3 y - 1) d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 y^{2} - 2 y$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $3 y^{2} - 2 y$ .

$\frac{d}{d y} [3 y^{2} - 2 y]$

Step 1.1.2

By the Sum Rule, the derivative of $3 y^{2} - 2 y$ with respect to $y$ is $\frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- 2 y]$ .

$\frac{d}{d y} [3 y^{2}] + \frac{d}{d y} [- 2 y]$

Step 1.1.3

Evaluate $\frac{d}{d y} [3 y^{2}]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $y$ , the derivative of $3 y^{2}$ with respect to $y$ is $3 \frac{d}{d y} [y^{2}]$ .

$3 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 2 y]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$3 (2 y) + \frac{d}{d y} [- 2 y]$

Step 1.1.3.3

Multiply $2$ by $3$ .

$6 y + \frac{d}{d y} [- 2 y]$

Step 1.1.4

Evaluate $\frac{d}{d y} [- 2 y]$ .

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Step 1.1.4.1

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2 y$ with respect to $y$ is $- 2 \frac{d}{d y} [y]$ .

$6 y - 2 \frac{d}{d y} [y]$

Step 1.1.4.2

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

$6 y - 2 \cdot 1$

Step 1.1.4.3

Multiply $- 2$ by $1$ .

$6 y - 2$

Step 1.2

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

$\int \sqrt{u} \frac{1}{2} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$\int \frac{\sqrt{u}}{2} d u$

Step 3

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

$\frac{1}{2} \int \sqrt{u} d u$

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Replace all occurrences of $u$ with $3 y^{2} - 2 y$ .

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