Calculus Examples

$2 (y + 3) d x - x (y d) y = 0$

Step 1

Subtract $2 (y + 3) d x$ from both sides of the equation.

$- x y d y = - 2 (y + 3) d x$

Step 2

Multiply both sides by $\frac{1}{x (y + 3)}$ .

$\frac{1}{x (y + 3)} (- x y) d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$- \frac{1}{x (y + 3)} (x y) d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{x (y + 3)}$ into the numerator.

$\frac{- 1}{x (y + 3)} (x y) d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.2.2

Factor $x$ out of $x y$ .

$\frac{- 1}{x (y + 3)} (x (y)) d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.2.3

Cancel the common factor.

$\frac{- 1}{x (y + 3)} (x y) d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.2.4

Rewrite the expression.

$\frac{- 1}{y + 3} y d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.3

Combine $\frac{- 1}{y + 3}$ and $y$ .

$\frac{- y}{y + 3} d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.4

Move the negative in front of the fraction.

$- \frac{y}{y + 3} d y = \frac{1}{x (y + 3)} (- 2 (y + 3)) d x$

Step 3.5

Rewrite using the commutative property of multiplication.

$- \frac{y}{y + 3} d y = - 2 \frac{1}{x (y + 3)} (y + 3) d x$

Step 3.6

Combine $- 2$ and $\frac{1}{x (y + 3)}$ .

$- \frac{y}{y + 3} d y = \frac{- 2}{x (y + 3)} (y + 3) d x$

Step 3.7

Cancel the common factor of $y + 3$ .

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Step 3.7.1

Factor $y + 3$ out of $x (y + 3)$ .

$- \frac{y}{y + 3} d y = \frac{- 2}{(y + 3) x} (y + 3) d x$

Step 3.7.2

Cancel the common factor.

$- \frac{y}{y + 3} d y = \frac{- 2}{(y + 3) x} (y + 3) d x$

Step 3.7.3

Rewrite the expression.

$- \frac{y}{y + 3} d y = \frac{- 2}{x} d x$

Step 3.8

Move the negative in front of the fraction.

$- \frac{y}{y + 3} d y = - \frac{2}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

Divide $y$ by $y + 3$ .

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Step 4.2.2.1

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

$y$

$3$

$y$

$0$

Step 4.2.2.2

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

					$1$
	$y$	+	$3$		$y$	+	$0$

Step 4.2.2.3

Multiply the new quotient term by the divisor.

				$1$
$y$	+	$3$		$y$	+	$0$
			+	$y$	+	$3$

Step 4.2.2.4

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

				$1$
$y$	+	$3$		$y$	+	$0$
			-	$y$	-	$3$

Step 4.2.2.5

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

				$1$
$y$	+	$3$		$y$	+	$0$
			-	$y$	-	$3$
					-	$3$

Step 4.2.2.6

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

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

Step 4.2.3

Split the single integral into multiple integrals.

$- (\int d y + \int - \frac{3}{y + 3} d y) = \int - \frac{2}{x} d x$

Step 4.2.4

Apply the constant rule.

$- (y + C_{1} + \int - \frac{3}{y + 3} d y) = \int - \frac{2}{x} d x$

Step 4.2.5

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

$- (y + C_{1} - \int \frac{3}{y + 3} d y) = \int - \frac{2}{x} d x$

Step 4.2.6

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$- (y + C_{1} - (3 \int \frac{1}{y + 3} d y)) = \int - \frac{2}{x} d x$

Step 4.2.7

Multiply $3$ by $- 1$ .

$- (y + C_{1} - 3 \int \frac{1}{y + 3} d y) = \int - \frac{2}{x} d x$

Step 4.2.8

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

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Step 4.2.8.1

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

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Step 4.2.8.1.1

Differentiate $y + 3$ .

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

Step 4.2.8.1.2

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 4.2.8.1.3

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 4.2.8.1.4

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

$1 + 0$

Step 4.2.8.1.5

Add $1$ and $0$ .

$1$

Step 4.2.8.2

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

$- (y + C_{1} - 3 \int \frac{1}{u} d u) = \int - \frac{2}{x} d x$

Step 4.2.9

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

$- (y + C_{1} - 3 (\ln (| u |) + C_{2})) = \int - \frac{2}{x} d x$

Step 4.2.10

Simplify.

$- (y - 3 \ln (| u |)) + C_{3} = \int - \frac{2}{x} d x$

Step 4.2.11

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

$- (y - 3 \ln (| y + 3 |)) + C_{3} = \int - \frac{2}{x} d x$

Step 4.2.12

Simplify.

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Step 4.2.12.1

Apply the distributive property.

$- y - (- 3 \ln (| y + 3 |)) + C_{3} = \int - \frac{2}{x} d x$

Step 4.2.12.2

Multiply $- 3$ by $- 1$ .

$- y + 3 \ln (| y + 3 |) + C_{3} = \int - \frac{2}{x} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

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

$- y + 3 \ln (| y + 3 |) + C_{3} = - \int \frac{2}{x} d x$

Step 4.3.2

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

$- y + 3 \ln (| y + 3 |) + C_{3} = - (2 \int \frac{1}{x} d x)$

Step 4.3.3

Multiply $2$ by $- 1$ .

$- y + 3 \ln (| y + 3 |) + C_{3} = - 2 \int \frac{1}{x} d x$

Step 4.3.4

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

$- y + 3 \ln (| y + 3 |) + C_{3} = - 2 (\ln (| x |) + C_{4})$

Step 4.3.5

Simplify.

$- y + 3 \ln (| y + 3 |) + C_{3} = - 2 \ln (| x |) + C_{4}$

Step 4.4

Group the constant of integration on the right side as $C$ .

$- y + 3 \ln (| y + 3 |) = - 2 \ln (| x |) + C$