Calculus Examples

$4 x y \frac{d y}{d x} = x^{2} + 1$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $4 x y \frac{d y}{d x} = x^{2} + 1$ by $4 x y$ and simplify.

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Step 1.1.1

Divide each term in $4 x y \frac{d y}{d x} = x^{2} + 1$ by $4 x y$ .

$\frac{4 x y \frac{d y}{d x}}{4 x y} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $4$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{4 x y \frac{d y}{d x}}{4 x y} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.1

Cancel the common factor.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.2.2.2

Rewrite the expression.

$\frac{y \frac{d y}{d x}}{y} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.2.3

Cancel the common factor of $y$ .

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Step 1.1.2.3.1

Cancel the common factor.

$\frac{y \frac{d y}{d x}}{y} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.2.3.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{x^{2}}{4 x y} + \frac{1}{4 x y}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.1.3.1.1

Factor $x$ out of $x^{2}$ .

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

Step 1.1.3.1.2

Cancel the common factors.

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Step 1.1.3.1.2.1

Factor $x$ out of $4 x y$ .

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

Step 1.1.3.1.2.2

Cancel the common factor.

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

Step 1.1.3.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{x}{4 y} + \frac{1}{4 x y}$

Step 1.2

Factor.

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Step 1.2.1

To write $\frac{x}{4 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2.2

Write each expression with a common denominator of $4 y x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

Multiply $\frac{x}{4 y}$ by $\frac{x}{x}$ .

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

Step 1.2.2.2

Reorder the factors of $4 y x$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Multiply $x$ by $x$ .

$\frac{d y}{d x} = \frac{x^{2} + 1}{4 x y}$

Step 1.3

Regroup factors.

$\frac{d y}{d x} = \frac{x^{2} + 1}{4 x} \cdot \frac{1}{y}$

Step 1.4

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (\frac{x^{2} + 1}{4 x} \cdot \frac{1}{y})$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $\frac{x^{2} + 1}{4 x}$ by $\frac{1}{y}$ .

$y \frac{d y}{d x} = y \frac{x^{2} + 1}{4 x y}$

Step 1.5.2

Cancel the common factor of $y$ .

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Step 1.5.2.1

Factor $y$ out of $4 x y$ .

$y \frac{d y}{d x} = y \frac{x^{2} + 1}{y (4 x)}$

Step 1.5.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x^{2} + 1}{y (4 x)}$

Step 1.5.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{x^{2} + 1}{4 x}$

Step 1.6

Rewrite the equation.

$y d y = \frac{x^{2} + 1}{4 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x^{2} + 1}{4 x} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x^{2} + 1}{4 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{x^{2} + 1}{x} d x$

Step 2.3.2

Divide $x^{2} + 1$ by $x$ .

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Step 2.3.2.1

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

$x$

$0$

$x^{2}$

$0 x$

$1$

Step 2.3.2.2

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

					$x$
	$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$

Step 2.3.2.3

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	+	$0$

Step 2.3.2.4

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

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

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int x + \frac{1}{x} d x$

Step 2.3.3

Split the single integral into multiple integrals.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} (\int x d x + \int \frac{1}{x} d x)$

Step 2.3.4

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} (\frac{1}{2} x^{2} + C_{2} + \int \frac{1}{x} d x)$

Step 2.3.5

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

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

Step 2.3.6

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C_{4}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (\frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{4} (\frac{1}{2} x^{2} + \ln (| x |)) + C)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

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

$y^{2} = 2 (\frac{1}{4} (\frac{x^{2}}{2} + \ln (| x |)) + C)$

Step 3.2.2.1.1.2

Apply the distributive property.

$y^{2} = 2 (\frac{1}{4} \cdot \frac{x^{2}}{2} + \frac{1}{4} \ln (| x |) + C)$

Step 3.2.2.1.1.3

Combine.

$y^{2} = 2 (\frac{1 x^{2}}{4 \cdot 2} + \frac{1}{4} \ln (| x |) + C)$

Step 3.2.2.1.1.4

Combine $\frac{1}{4}$ and $\ln (| x |)$ .

$y^{2} = 2 (\frac{1 x^{2}}{4 \cdot 2} + \frac{\ln (| x |)}{4} + C)$

Step 3.2.2.1.1.5

Simplify each term.

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Step 3.2.2.1.1.5.1

Multiply $x^{2}$ by $1$ .

$y^{2} = 2 (\frac{x^{2}}{4 \cdot 2} + \frac{\ln (| x |)}{4} + C)$

Step 3.2.2.1.1.5.2

Multiply $4$ by $2$ .

$y^{2} = 2 (\frac{x^{2}}{8} + \frac{\ln (| x |)}{4} + C)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{x^{2}}{8} + 2 \frac{\ln (| x |)}{4} + 2 C$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1.1

Factor $2$ out of $8$ .

$y^{2} = 2 \frac{x^{2}}{2 (4)} + 2 \frac{\ln (| x |)}{4} + 2 C$

Step 3.2.2.1.3.1.2

Cancel the common factor.

$y^{2} = 2 \frac{x^{2}}{2 \cdot 4} + 2 \frac{\ln (| x |)}{4} + 2 C$

Step 3.2.2.1.3.1.3

Rewrite the expression.

$y^{2} = \frac{x^{2}}{4} + 2 \frac{\ln (| x |)}{4} + 2 C$

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

Factor $2$ out of $4$ .

$y^{2} = \frac{x^{2}}{4} + 2 \frac{\ln (| x |)}{2 (2)} + 2 C$

Step 3.2.2.1.3.2.2

Cancel the common factor.

$y^{2} = \frac{x^{2}}{4} + 2 \frac{\ln (| x |)}{2 \cdot 2} + 2 C$

Step 3.2.2.1.3.2.3

Rewrite the expression.

$y^{2} = \frac{x^{2}}{4} + \frac{\ln (| x |)}{2} + 2 C$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{x^{2}}{4} + \frac{\ln (| x |)}{2} + 2 C}$

Step 3.4

Simplify $\pm \sqrt{\frac{x^{2}}{4} + \frac{\ln (| x |)}{2} + 2 C}$ .

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Rewrite $\frac{\ln (| x |)}{2}$ as $\frac{1}{2} \ln (| x |)$ .

$y = \pm \sqrt{\frac{x^{2}}{4} + \frac{1}{2} \ln (| x |) + 2 C}$

Step 3.4.1.2

Simplify $\frac{1}{2} \ln (| x |)$ by moving $\frac{1}{2}$ inside the logarithm.

$y = \pm \sqrt{\frac{x^{2}}{4} + \ln ({| x |}^{\frac{1}{2}}) + 2 C}$

Step 3.4.2

To write $\ln ({| x |}^{\frac{1}{2}})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{x^{2}}{4} + \ln ({| x |}^{\frac{1}{2}}) \cdot \frac{4}{4} + 2 C}$

Step 3.4.3

Simplify terms.

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Step 3.4.3.1

Combine $\ln ({| x |}^{\frac{1}{2}})$ and $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{x^{2}}{4} + \frac{\ln ({| x |}^{\frac{1}{2}}) \cdot 4}{4} + 2 C}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} + \ln ({| x |}^{\frac{1}{2}}) \cdot 4}{4} + 2 C}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Multiply $\ln ({| x |}^{\frac{1}{2}}) \cdot 4$ .

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Step 3.4.4.1.1

Reorder $\ln ({| x |}^{\frac{1}{2}})$ and $4$ .

$y = \pm \sqrt{\frac{x^{2} + 4 \cdot \ln ({| x |}^{\frac{1}{2}})}{4} + 2 C}$

Step 3.4.4.1.2

Simplify $4 \cdot \ln ({| x |}^{\frac{1}{2}})$ by moving $4$ inside the logarithm.

$y = \pm \sqrt{\frac{x^{2} + \ln ({({| x |}^{\frac{1}{2}})}^{4})}{4} + 2 C}$

Step 3.4.4.2

Multiply the exponents in ${({| x |}^{\frac{1}{2}})}^{4}$ .

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Step 3.4.4.2.1

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

$y = \pm \sqrt{\frac{x^{2} + \ln ({| x |}^{\frac{1}{2} \cdot 4})}{4} + 2 C}$

Step 3.4.4.2.2

Cancel the common factor of $2$ .

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Step 3.4.4.2.2.1

Factor $2$ out of $4$ .

$y = \pm \sqrt{\frac{x^{2} + \ln ({| x |}^{\frac{1}{2} \cdot (2 (2))})}{4} + 2 C}$

Step 3.4.4.2.2.2

Cancel the common factor.

$y = \pm \sqrt{\frac{x^{2} + \ln ({| x |}^{\frac{1}{2} \cdot (2 \cdot 2)})}{4} + 2 C}$

Step 3.4.4.2.2.3

Rewrite the expression.

$y = \pm \sqrt{\frac{x^{2} + \ln ({| x |}^{2})}{4} + 2 C}$

Step 3.4.4.3

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$y = \pm \sqrt{\frac{x^{2} + \ln (x^{2})}{4} + 2 C}$

Step 3.4.5

To write $2 C$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{x^{2} + \ln (x^{2})}{4} + 2 C \cdot \frac{4}{4}}$

Step 3.4.6

Combine $2 C$ and $\frac{4}{4}$ .

$y = \pm \sqrt{\frac{x^{2} + \ln (x^{2})}{4} + \frac{2 C \cdot 4}{4}}$

Step 3.4.7

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} + \ln (x^{2}) + 2 C \cdot 4}{4}}$

Step 3.4.8

Multiply $4$ by $2$ .

$y = \pm \sqrt{\frac{x^{2} + \ln (x^{2}) + 8 C}{4}}$

Step 3.4.9

Rewrite $\frac{x^{2} + \ln (x^{2}) + 8 C}{4}$ as ${(\frac{1}{2})}^{2} (x^{2} + \ln (x^{2}) + 8 C)$ .

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Step 3.4.9.1

Factor the perfect power $1^{2}$ out of $x^{2} + \ln (x^{2}) + 8 C$ .

$y = \pm \sqrt{\frac{1^{2} (x^{2} + \ln (x^{2}) + 8 C)}{4}}$

Step 3.4.9.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{\frac{1^{2} (x^{2} + \ln (x^{2}) + 8 C)}{2^{2} \cdot 1}}$

Step 3.4.9.3

Rearrange the fraction $\frac{1^{2} (x^{2} + \ln (x^{2}) + 8 C)}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} (x^{2} + \ln (x^{2}) + 8 C)}$

Step 3.4.10

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{x^{2} + \ln (x^{2}) + 8 C}$

Step 3.4.11

Combine $\frac{1}{2}$ and $\sqrt{x^{2} + \ln (x^{2}) + 8 C}$ .

$y = \pm \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

$y = - \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

$y = \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

$y = - \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

$y = \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

$y = - \frac{\sqrt{x^{2} + \ln (x^{2}) + 8 C}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{x^{2} + \ln (x^{2}) + C}}{2}$

$y = - \frac{\sqrt{x^{2} + \ln (x^{2}) + C}}{2}$