Calculus Examples

$\frac{d y}{d x} = \frac{4 y}{x (y - 3)}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{4 y}{y - 3} \cdot \frac{1}{x}$

Step 1.2

Multiply both sides by $\frac{y - 3}{4 y}$ .

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{y - 3}{4 y} (\frac{4 y}{y - 3} \cdot \frac{1}{x})$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $\frac{4 y}{y - 3}$ by $\frac{1}{x}$ .

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{y - 3}{4 y} \cdot \frac{4 y}{(y - 3) x}$

Step 1.3.2

Cancel the common factor of $y - 3$ .

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Step 1.3.2.1

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

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{y - 3}{4 y} \cdot \frac{4 y}{(y - 3) (x)}$

Step 1.3.2.2

Cancel the common factor.

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{y - 3}{4 y} \cdot \frac{4 y}{(y - 3) x}$

Step 1.3.2.3

Rewrite the expression.

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{1}{4 y} \cdot \frac{4 y}{x}$

Step 1.3.3

Cancel the common factor of $4 y$ .

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Step 1.3.3.1

Cancel the common factor.

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{1}{4 y} \cdot \frac{4 y}{x}$

Step 1.3.3.2

Rewrite the expression.

$\frac{y - 3}{4 y} \frac{d y}{d x} = \frac{1}{x}$

Step 1.4

Rewrite the equation.

$\frac{y - 3}{4 y} d y = \frac{1}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{y - 3}{4 y} d y = \int \frac{1}{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $\frac{1}{4}$ is constant with respect to $y$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{y - 3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.2

Divide $y - 3$ by $y$ .

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Step 2.2.2.1

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

$y$

$0$

$y$

$3$

Step 2.2.2.2

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

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

Step 2.2.2.3

Multiply the new quotient term by the divisor.

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

Step 2.2.2.4

The expression needs to be subtracted from the dividend, so change all the signs in $y + 0$

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

Step 2.2.2.5

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

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

Step 2.2.2.6

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

$\frac{1}{4} \int 1 - \frac{3}{y} d y = \int \frac{1}{x} d x$

Step 2.2.3

Split the single integral into multiple integrals.

$\frac{1}{4} (\int d y + \int - \frac{3}{y} d y) = \int \frac{1}{x} d x$

Step 2.2.4

Apply the constant rule.

$\frac{1}{4} (y + C_{1} + \int - \frac{3}{y} d y) = \int \frac{1}{x} d x$

Step 2.2.5

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

$\frac{1}{4} (y + C_{1} - \int \frac{3}{y} d y) = \int \frac{1}{x} d x$

Step 2.2.6

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

$\frac{1}{4} (y + C_{1} - (3 \int \frac{1}{y} d y)) = \int \frac{1}{x} d x$

Step 2.2.7

Multiply $3$ by $- 1$ .

$\frac{1}{4} (y + C_{1} - 3 \int \frac{1}{y} d y) = \int \frac{1}{x} d x$

Step 2.2.8

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

$\frac{1}{4} (y + C_{1} - 3 (\ln (| y |) + C_{2})) = \int \frac{1}{x} d x$

Step 2.2.9

Simplify.

$\frac{1}{4} (y - 3 \ln (| y |)) + C_{3} = \int \frac{1}{x} d x$

Step 2.3

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

$\frac{1}{4} (y - 3 \ln (| y |)) + C_{3} = \ln (| x |) + C_{4}$

Step 2.4

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

$\frac{1}{4} (y - 3 \ln (| y |)) = \ln (| x |) + C$