Calculus Examples

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

Step 1

Divide $y - 2$ by $y + 3$ .

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Step 1.1

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

$y$

$3$

$y$

$2$

Step 1.2

Divide the highest order term in the dividend $y$ by the highest order term in divisor $y$ .

					$1$
	$y$	+	$3$		$y$	-	$2$

Step 1.3

Multiply the new quotient term by the divisor.

				$1$
$y$	+	$3$		$y$	-	$2$
			+	$y$	+	$3$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $y + 3$

				$1$
$y$	+	$3$		$y$	-	$2$
			-	$y$	-	$3$

Step 1.5

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

				$1$
$y$	+	$3$		$y$	-	$2$
			-	$y$	-	$3$
					-	$5$

Step 1.6

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

$\int 1 - \frac{5}{y + 3} d y$

Step 2

Split the single integral into multiple integrals.

$\int d y + \int - \frac{5}{y + 3} d y$

Step 3

Apply the constant rule.

$y + C + \int - \frac{5}{y + 3} d y$

Step 4

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

$y + C - \int \frac{5}{y + 3} d y$

Step 5

Since $5$ is constant with respect to $y$ , move $5$ out of the integral.

$y + C - (5 \int \frac{1}{y + 3} d y)$

Step 6

Multiply $5$ by $- 1$ .

$y + C - 5 \int \frac{1}{y + 3} d y$

Step 7

Let $u = y + 3$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = y + 3$ . Find $\frac{d u}{d y}$ .

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Step 7.1.1

Differentiate $y + 3$ .

$\frac{d}{d y} [y + 3]$

Step 7.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [3]$

Step 7.1.3

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

$1 + \frac{d}{d y} [3]$

Step 7.1.4

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$1 + 0$

Step 7.1.5

Add $1$ and $0$ .

$1$

Step 7.2

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

$y + C - 5 \int \frac{1}{u} d u$

Step 8

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

$y + C - 5 (\ln (| u |) + C)$

Step 9

Simplify.

$y - 5 \ln (| u |) + C$

Step 10

Replace all occurrences of $u$ with $y + 3$ .

$y - 5 \ln (| y + 3 |) + C$