Calculus Examples

$\int \frac{1}{b y^{2} + b^{3}} d y$

Step 1

Factor $b$ out of $b y^{2} + b^{3}$ .

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Step 1.1

Factor $b$ out of $b y^{2}$ .

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

Step 1.2

Factor $b$ out of $b^{3}$ .

$\int \frac{1}{b (y^{2}) + b \cdot b^{2}} d y$

Step 1.3

Factor $b$ out of $b (y^{2}) + b \cdot b^{2}$ .

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

Step 2

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

$\frac{1}{b} \int \frac{1}{y^{2} + b^{2}} d y$

Step 3

Reorder $y^{2}$ and $b^{2}$ .

$\frac{1}{b} \int \frac{1}{b^{2} + y^{2}} d y$

Step 4

The integral of $\frac{1}{b^{2} + y^{2}}$ with respect to $y$ is $\frac{1}{b} \arctan (\frac{y}{b}) + C$ .

$\frac{1}{b} (\frac{1}{b} \arctan (\frac{y}{b}) + C)$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{b}$ and $\arctan (\frac{y}{b})$ .

$\frac{1}{b} (\frac{\arctan (\frac{y}{b})}{b} + C)$

Step 5.2

Simplify.

$\frac{1}{b} \cdot \frac{\arctan (\frac{y}{b})}{b} + C$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $\frac{1}{b}$ by $\frac{\arctan (\frac{y}{b})}{b}$ .

$\frac{\arctan (\frac{y}{b})}{b \cdot b} + C$

Step 5.3.2

Raise $b$ to the power of $1$ .

$\frac{\arctan (\frac{y}{b})}{b^{1} b} + C$

Step 5.3.3

Raise $b$ to the power of $1$ .

$\frac{\arctan (\frac{y}{b})}{b^{1} b^{1}} + C$

Step 5.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\arctan (\frac{y}{b})}{b^{1 + 1}} + C$

Step 5.3.5

Add $1$ and $1$ .

$\frac{\arctan (\frac{y}{b})}{b^{2}} + C$