Calculus Examples

$2 y (x^{2} + 1) d x + (\frac{x^{3}}{3} + x) d y = 0$

Step 1

Subtract $2 y (x^{2} + 1) d x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $\frac{x^{3}}{3} + x$ .

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Step 3.1.1

Factor $\frac{x^{3}}{3} + x$ out of $(\frac{x^{3}}{3} + x) y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

Factor $x$ out of $\frac{x^{3}}{3} + x$ .

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Step 3.3.1.1

Factor $x$ out of $\frac{x^{3}}{3}$ .

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

Step 3.3.1.2

Raise $x$ to the power of $1$ .

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

Step 3.3.1.3

Factor $x$ out of $x^{1}$ .

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

Step 3.3.1.4

Factor $x$ out of $x \frac{x^{2}}{3} + x \cdot 1$ .

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

Step 3.3.2

Write $1$ as a fraction with a common denominator.

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

Step 3.3.3

Combine the numerators over the common denominator.

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

Step 3.3.4

Combine exponents.

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Step 3.3.4.1

Combine $x$ and $\frac{x^{2} + 3}{3}$ .

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

Step 3.3.4.2

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply $\frac{3}{x (x^{2} + 3) y}$ by $1$ .

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

Step 3.6

Multiply $- 2 \frac{3}{x (x^{2} + 3) y}$ .

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Step 3.6.1

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

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

Step 3.6.2

Multiply $- 2$ by $3$ .

$\frac{1}{y} d y = \frac{- 6}{x (x^{2} + 3) y} (y (x^{2} + 1)) d x$

Step 3.7

Cancel the common factor of $y$ .

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Step 3.7.1

Factor $y$ out of $x (x^{2} + 3) y$ .

$\frac{1}{y} d y = \frac{- 6}{y (x (x^{2} + 3))} (y (x^{2} + 1)) d x$

Step 3.7.2

Cancel the common factor.

$\frac{1}{y} d y = \frac{- 6}{y (x (x^{2} + 3))} (y (x^{2} + 1)) d x$

Step 3.7.3

Rewrite the expression.

$\frac{1}{y} d y = \frac{- 6}{x (x^{2} + 3)} (x^{2} + 1) d x$

Step 3.8

Move the negative in front of the fraction.

$\frac{1}{y} d y = - \frac{6}{x (x^{2} + 3)} (x^{2} + 1) d x$

Step 3.9

Apply the distributive property.

$\frac{1}{y} d y = (- \frac{6}{x (x^{2} + 3)} x^{2} - \frac{6}{x (x^{2} + 3)} \cdot 1) d x$

Step 3.10

Cancel the common factor of $x$ .

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Step 3.10.1

Move the leading negative in $- \frac{6}{x (x^{2} + 3)}$ into the numerator.

$\frac{1}{y} d y = (\frac{- 6}{x (x^{2} + 3)} x^{2} - \frac{6}{x (x^{2} + 3)} \cdot 1) d x$

Step 3.10.2

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

$\frac{1}{y} d y = (\frac{- 6}{x (x^{2} + 3)} (x \cdot x) - \frac{6}{x (x^{2} + 3)} \cdot 1) d x$

Step 3.10.3

Cancel the common factor.

$\frac{1}{y} d y = (\frac{- 6}{x (x^{2} + 3)} (x \cdot x) - \frac{6}{x (x^{2} + 3)} \cdot 1) d x$

Step 3.10.4

Rewrite the expression.

$\frac{1}{y} d y = (\frac{- 6}{x^{2} + 3} x - \frac{6}{x (x^{2} + 3)} \cdot 1) d x$

Step 3.11

Combine $\frac{- 6}{x^{2} + 3}$ and $x$ .

$\frac{1}{y} d y = (\frac{- 6 x}{x^{2} + 3} - \frac{6}{x (x^{2} + 3)} \cdot 1) d x$

Step 3.12

Multiply $- 1$ by $1$ .

$\frac{1}{y} d y = (\frac{- 6 x}{x^{2} + 3} - \frac{6}{x (x^{2} + 3)}) d x$

Step 3.13

Move the negative in front of the fraction.

$\frac{1}{y} d y = (- \frac{6 x}{x^{2} + 3} - \frac{6}{x (x^{2} + 3)}) d x$

Step 3.14

To write $- \frac{6 x}{x^{2} + 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{y} d y = (- \frac{6 x}{x^{2} + 3} \cdot \frac{x}{x} - \frac{6}{x (x^{2} + 3)}) d x$

Step 3.15

Write each expression with a common denominator of $(x^{2} + 3) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.15.1

Multiply $\frac{6 x}{x^{2} + 3}$ by $\frac{x}{x}$ .

$\frac{1}{y} d y = (- \frac{6 x \cdot x}{(x^{2} + 3) x} - \frac{6}{x (x^{2} + 3)}) d x$

Step 3.15.2

Reorder the factors of $(x^{2} + 3) x$ .

$\frac{1}{y} d y = (- \frac{6 x \cdot x}{x (x^{2} + 3)} - \frac{6}{x (x^{2} + 3)}) d x$

Step 3.16

Combine the numerators over the common denominator.

$\frac{1}{y} d y = \frac{- 6 x \cdot x - 6}{x (x^{2} + 3)} d x$

Step 3.17

Simplify the numerator.

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Step 3.17.1

Factor $6$ out of $- 6 x \cdot x - 6$ .

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Step 3.17.1.1

Factor $6$ out of $- 6 x \cdot x$ .

$\frac{1}{y} d y = \frac{6 (- x \cdot x) - 6}{x (x^{2} + 3)} d x$

Step 3.17.1.2

Factor $6$ out of $- 6$ .

$\frac{1}{y} d y = \frac{6 (- x \cdot x) + 6 (- 1)}{x (x^{2} + 3)} d x$

Step 3.17.1.3

Factor $6$ out of $6 (- x \cdot x) + 6 (- 1)$ .

$\frac{1}{y} d y = \frac{6 (- x \cdot x - 1)}{x (x^{2} + 3)} d x$

Step 3.17.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.17.2.1

Move $x$ .

$\frac{1}{y} d y = \frac{6 (- (x \cdot x) - 1)}{x (x^{2} + 3)} d x$

Step 3.17.2.2

Multiply $x$ by $x$ .

$\frac{1}{y} d y = \frac{6 (- x^{2} - 1)}{x (x^{2} + 3)} d x$

Step 3.18

Factor $- 1$ out of $- x^{2}$ .

$\frac{1}{y} d y = \frac{6 (- (x^{2}) - 1)}{x (x^{2} + 3)} d x$

Step 3.19

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{1}{y} d y = \frac{6 (- (x^{2}) - 1 (1))}{x (x^{2} + 3)} d x$

Step 3.20

Factor $- 1$ out of $- (x^{2}) - 1 (1)$ .

$\frac{1}{y} d y = \frac{6 (- (x^{2} + 1))}{x (x^{2} + 3)} d x$

Step 3.21

Rewrite $- (x^{2} + 1)$ as $- 1 (x^{2} + 1)$ .

$\frac{1}{y} d y = \frac{6 (- 1 (x^{2} + 1))}{x (x^{2} + 3)} d x$

Step 3.22

Move the negative in front of the fraction.

$\frac{1}{y} d y = - \frac{6 (x^{2} + 1)}{x (x^{2} + 3)} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int - \frac{6 (x^{2} + 1)}{x (x^{2} + 3)} d x$

Step 4.2

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

$\ln (| y |) + C_{1} = \int - \frac{6 (x^{2} + 1)}{x (x^{2} + 3)} d x$

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.

$\ln (| y |) + C_{1} = - \int \frac{6 (x^{2} + 1)}{x (x^{2} + 3)} d x$

Step 4.3.2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\ln (| y |) + C_{1} = - (6 \int \frac{x^{2} + 1}{x (x^{2} + 3)} d x)$

Step 4.3.3

Multiply $6$ by $- 1$ .

$\ln (| y |) + C_{1} = - 6 \int \frac{x^{2} + 1}{x (x^{2} + 3)} d x$

Step 4.3.4

Let $u = x (x^{2} + 3)$ . Then $d u = (3 x^{2} + 3) d x$ , so $\frac{1}{3} d u = (x^{2} + 1) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.4.1

Let $u = x (x^{2} + 3)$ . Find $\frac{d u}{d x}$ .

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Step 4.3.4.1.1

Differentiate $x (x^{2} + 3)$ .

$\frac{d}{d x} [x (x^{2} + 3)]$

Step 4.3.4.1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} + 3$ .

$x \frac{d}{d x} [x^{2} + 3] + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.3

Differentiate.

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Step 4.3.4.1.3.1

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]) + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x (2 x + \frac{d}{d x} [3]) + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.3.3

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

$x (2 x + 0) + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.3.4

Add $2 x$ and $0$ .

$x (2 x) + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.4

Raise $x$ to the power of $1$ .

$2 (x^{1} x) + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.5

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{1}) + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.7

Add $1$ and $1$ .

$2 x^{2} + (x^{2} + 3) \frac{d}{d x} [x]$

Step 4.3.4.1.8

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

$2 x^{2} + (x^{2} + 3) \cdot 1$

Step 4.3.4.1.9

Simplify by adding terms.

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Step 4.3.4.1.9.1

Multiply $x^{2} + 3$ by $1$ .

$2 x^{2} + x^{2} + 3$

Step 4.3.4.1.9.2

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} + 3$

Step 4.3.4.2

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

$\ln (| y |) + C_{1} = - 6 \int \frac{1}{u} \cdot \frac{1}{3} d u$

Step 4.3.5

Simplify.

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Step 4.3.5.1

Multiply $\frac{1}{u}$ by $\frac{1}{3}$ .

$\ln (| y |) + C_{1} = - 6 \int \frac{1}{u \cdot 3} d u$

Step 4.3.5.2

Move $3$ to the left of $u$ .

$\ln (| y |) + C_{1} = - 6 \int \frac{1}{3 u} d u$

Step 4.3.6

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

$\ln (| y |) + C_{1} = - 6 (\frac{1}{3} \int \frac{1}{u} d u)$

Step 4.3.7

Simplify.

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Step 4.3.7.1

Combine $\frac{1}{3}$ and $- 6$ .

$\ln (| y |) + C_{1} = \frac{- 6}{3} \int \frac{1}{u} d u$

Step 4.3.7.2

Cancel the common factor of $- 6$ and $3$ .

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Step 4.3.7.2.1

Factor $3$ out of $- 6$ .

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

Step 4.3.7.2.2

Cancel the common factors.

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Step 4.3.7.2.2.1

Factor $3$ out of $3$ .

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

Step 4.3.7.2.2.2

Cancel the common factor.

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

Step 4.3.7.2.2.3

Rewrite the expression.

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

Step 4.3.7.2.2.4

Divide $- 2$ by $1$ .

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

Step 4.3.8

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

$\ln (| y |) + C_{1} = - 2 (\ln (| u |) + C_{2})$

Step 4.3.9

Simplify.

$\ln (| y |) + C_{1} = - 2 \ln (| u |) + C_{2}$

Step 4.3.10

Replace all occurrences of $u$ with $x (x^{2} + 3)$ .

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| y |) + 2 \ln (| x (x^{2} + 3) |) = C$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$\ln (| y |) + 2 \ln (| x \cdot x^{2} + x \cdot 3 |) = C$

Step 5.2.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Raise $x$ to the power of $1$ .

$\ln (| y |) + 2 \ln (| x \cdot x^{2} + x \cdot 3 |) = C$

Step 5.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (| y |) + 2 \ln (| x^{1 + 2} + x \cdot 3 |) = C$

Step 5.2.2.2

Add $1$ and $2$ .

$\ln (| y |) + 2 \ln (| x^{3} + x \cdot 3 |) = C$

Step 5.2.3

Move $3$ to the left of $x$ .

$\ln (| y |) + 2 \ln (| x^{3} + 3 x |) = C$

Step 5.3

Simplify the left side.

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Step 5.3.1

Simplify $\ln (| y |) + 2 \ln (| x^{3} + 3 x |)$ .

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Step 5.3.1.1

Simplify each term.

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Step 5.3.1.1.1

Simplify $2 \ln (| x^{3} + 3 x |)$ by moving $2$ inside the logarithm.

$\ln (| y |) + \ln ({| x^{3} + 3 x |}^{2}) = C$

Step 5.3.1.1.2

Remove the absolute value in ${| x^{3} + 3 x |}^{2}$ because exponentiations with even powers are always positive.

$\ln (| y |) + \ln ({(x^{3} + 3 x)}^{2}) = C$

Step 5.3.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (| y | {(x^{3} + 3 x)}^{2}) = C$

Step 5.4

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y | {(x^{3} + 3 x)}^{2})} = e^{C}$

Step 5.5

Rewrite $\ln (| y | {(x^{3} + 3 x)}^{2}) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = | y | {(x^{3} + 3 x)}^{2}$

Step 5.6

Solve for $y$ .

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Step 5.6.1

Rewrite the equation as $| y | {(x^{3} + 3 x)}^{2} = e^{C}$ .

$| y | {(x^{3} + 3 x)}^{2} = e^{C}$

Step 5.6.2

Divide each term in $| y | {(x^{3} + 3 x)}^{2} = e^{C}$ by ${(x^{3} + 3 x)}^{2}$ and simplify.

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Step 5.6.2.1

Divide each term in $| y | {(x^{3} + 3 x)}^{2} = e^{C}$ by ${(x^{3} + 3 x)}^{2}$ .

$\frac{| y | {(x^{3} + 3 x)}^{2}}{{(x^{3} + 3 x)}^{2}} = \frac{e^{C}}{{(x^{3} + 3 x)}^{2}}$

Step 5.6.2.2

Simplify the left side.

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Step 5.6.2.2.1

Cancel the common factor of ${(x^{3} + 3 x)}^{2}$ .

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Step 5.6.2.2.1.1

Cancel the common factor.

$\frac{| y | {(x^{3} + 3 x)}^{2}}{{(x^{3} + 3 x)}^{2}} = \frac{e^{C}}{{(x^{3} + 3 x)}^{2}}$

Step 5.6.2.2.1.2

Divide $| y |$ by $1$ .

$| y | = \frac{e^{C}}{{(x^{3} + 3 x)}^{2}}$

Step 5.6.2.3

Simplify the right side.

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Step 5.6.2.3.1

Simplify the denominator.

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Step 5.6.2.3.1.1

Factor $x$ out of $x^{3} + 3 x$ .

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Step 5.6.2.3.1.1.1

Factor $x$ out of $x^{3}$ .

$| y | = \frac{e^{C}}{{(x \cdot x^{2} + 3 x)}^{2}}$

Step 5.6.2.3.1.1.2

Factor $x$ out of $3 x$ .

$| y | = \frac{e^{C}}{{(x \cdot x^{2} + x \cdot 3)}^{2}}$

Step 5.6.2.3.1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot 3$ .

$| y | = \frac{e^{C}}{{(x (x^{2} + 3))}^{2}}$

Step 5.6.2.3.1.2

Apply the product rule to $x (x^{2} + 3)$ .

$| y | = \frac{e^{C}}{x^{2} {(x^{2} + 3)}^{2}}$

Step 5.6.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm \frac{e^{C}}{x^{2} {(x^{2} + 3)}^{2}}$

Step 6

Group the constant terms together.

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Step 6.1

Simplify the constant of integration.

$y = \pm \frac{C}{x^{2} {(x^{2} + 3)}^{2}}$

Step 6.2

Combine constants with the plus or minus.

$y = C \frac{1}{x^{2} {(x^{2} + 3)}^{2}}$