Calculus Examples

$\int y (a - b y^{2}) d y$

Step 1

Simplify by multiplying through.

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Step 1.1

Expand $y (a - b y^{2})$ .

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Step 1.1.1

Apply the distributive property.

$\int y a + y (- b y^{2}) d y$

Step 1.1.2

Move parentheses.

$\int y a + y (- b) y^{2} d y$

Step 1.1.3

Reorder $y$ and $- 1$ .

$\int y a - 1 y b y^{2} d y$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite $- 1 y$ as $- y$ .

$\int y a - y b y^{2} d y$

Step 1.2.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 1.2.2.1

Move $y^{2}$ .

$\int y a - (y^{2} y) b d y$

Step 1.2.2.2

Multiply $y^{2}$ by $y$ .

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Step 1.2.2.2.1

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

$\int y a - (y^{2} y^{1}) b d y$

Step 1.2.2.2.2

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

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

Step 1.2.2.3

Add $2$ and $1$ .

$\int y a - y^{3} b d y$

Step 2

Split the single integral into multiple integrals.

$\int y a d y + \int - y^{3} b d y$

Step 3

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

$a \int y d y + \int - y^{3} b d y$

Step 4

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

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

Step 5

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

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

Step 6

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$a (\frac{1}{2} y^{2} + C) - b (\frac{1}{4} y^{4} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$\frac{1}{2} a y^{2} - b (\frac{1}{4} y^{4}) + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{1}{2}$ and $a$ .

$\frac{a}{2} y^{2} - b (\frac{1}{4} y^{4}) + C$

Step 7.2.2

Combine $\frac{a}{2}$ and $y^{2}$ .

$\frac{a y^{2}}{2} - b (\frac{1}{4} y^{4}) + C$

Step 7.2.3

Combine $\frac{1}{4}$ and $y^{4}$ .

$\frac{a y^{2}}{2} - b \frac{y^{4}}{4} + C$

Step 7.2.4

Combine $\frac{y^{4}}{4}$ and $b$ .

$\frac{a y^{2}}{2} - \frac{y^{4} b}{4} + C$

Step 8

Remove parentheses.

$\frac{1}{2} a y^{2} - \frac{1}{4} y^{4} b + C$