Calculus Examples

$\frac{y^{5} - 2 y}{y^{3}}$

Step 1

Simplify the expression.

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Step 1.1

Simplify.

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Step 1.1.1

Factor $y$ out of $y^{5} - 2 y$ .

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Step 1.1.1.1

Factor $y$ out of $y^{5}$ .

$\int \frac{y \cdot y^{4} - 2 y}{y^{3}} d y$

Step 1.1.1.2

Factor $y$ out of $- 2 y$ .

$\int \frac{y \cdot y^{4} + y \cdot - 2}{y^{3}} d y$

Step 1.1.1.3

Factor $y$ out of $y \cdot y^{4} + y \cdot - 2$ .

$\int \frac{y (y^{4} - 2)}{y^{3}} d y$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

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

$\int \frac{y (y^{4} - 2)}{y \cdot y^{2}} d y$

Step 1.1.2.2

Cancel the common factor.

$\int \frac{y (y^{4} - 2)}{y \cdot y^{2}} d y$

Step 1.1.2.3

Rewrite the expression.

$\int \frac{y^{4} - 2}{y^{2}} d y$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

Move $y^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int (y^{4} - 2) {(y^{2})}^{- 1} d y$

Step 1.2.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

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Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int (y^{4} - 2) y^{2 \cdot - 1} d y$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$\int (y^{4} - 2) y^{- 2} d y$

Step 2

Multiply $(y^{4} - 2) y^{- 2}$ .

$\int y^{4} y^{- 2} - 2 y^{- 2} d y$

Step 3

Multiply $y^{4}$ by $y^{- 2}$ by adding the exponents.

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Step 3.1

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

$\int y^{4 - 2} - 2 y^{- 2} d y$

Step 3.2

Subtract $2$ from $4$ .

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

Step 4

Split the single integral into multiple integrals.

$\int y^{2} d y + \int - 2 y^{- 2} d y$

Step 5

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

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

Step 6

Since $- 2$ is constant with respect to $y$ , move $- 2$ out of the integral.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Multiply $- 1$ by $- 2$ .

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