Calculus Examples

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

Step 1

Since $\frac{1}{3}$ is constant with respect to $y$ , move $\frac{1}{3}$ out of the integral.

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

Step 2

Reorder $1$ and $- y^{2}$ .

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

Step 3

Divide $- y^{2} + 1$ by $y$ .

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Step 3.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$y$

$0$

$y^{2}$

$0 y$

$1$

Step 3.2

Divide the highest order term in the dividend $- y^{2}$ by the highest order term in divisor $y$ .

				-	$y$
	$y$	+	$0$	-	$y^{2}$	+	$0 y$	+	$1$

Step 3.3

Multiply the new quotient term by the divisor.

			-	$y$
$y$	+	$0$	-	$y^{2}$	+	$0 y$	+	$1$
			-	$y^{2}$	+	$0$

Step 3.4

The expression needs to be subtracted from the dividend, so change all the signs in $- y^{2} + 0$

			-	$y$
$y$	+	$0$	-	$y^{2}$	+	$0 y$	+	$1$
			+	$y^{2}$	-	$0$

Step 3.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$y$
$y$	+	$0$	-	$y^{2}$	+	$0 y$	+	$1$
			+	$y^{2}$	-	$0$
						$0$

Step 3.6

Pull the next term from the original dividend down into the current dividend.

			-	$y$
$y$	+	$0$	-	$y^{2}$	+	$0 y$	+	$1$
			+	$y^{2}$	-	$0$
						$0$	+	$1$

Step 3.7

The final answer is the quotient plus the remainder over the divisor.

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

Step 4

Split the single integral into multiple integrals.

$\frac{1}{3} (\int - y d y + \int \frac{1}{y} d y)$

Step 5

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$\frac{1}{3} (- \int y d y + \int \frac{1}{y} d y)$

Step 6

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

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

Step 7

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

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

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $y^{2}$ .

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

Step 8.2

Simplify.

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