Calculus Examples

$y (u^{2} + 1) d u + (u^{3} - 3 u) d y = 0$

Step 1

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

$(u^{3} - 3 u) d y = - y (u^{2} + 1) d u$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $u^{3} - 3 u$ .

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Step 3.1.1

Factor $u^{3} - 3 u$ out of $(u^{3} - 3 u) y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

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

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

Step 3.3.2

Factor $y$ out of $(u^{3} - 3 u) y$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Factor $u$ out of $u^{3} - 3 u$ .

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Step 3.4.1

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

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

Step 3.4.2

Factor $u$ out of $- 3 u$ .

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

Step 3.4.3

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

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

Step 3.5

Move the negative in front of the fraction.

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Cancel the common factor of $u$ .

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Step 3.7.1

Move the leading negative in $- \frac{1}{u (u^{2} - 3)}$ into the numerator.

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

Step 3.7.2

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

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

Step 3.7.3

Cancel the common factor.

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

Step 3.7.4

Rewrite the expression.

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

Step 3.8

Combine $\frac{- 1}{u^{2} - 3}$ and $u$ .

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

Step 3.9

Multiply $- 1$ by $1$ .

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

Step 3.10

Move the negative in front of the fraction.

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

Step 3.11

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

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

Step 3.12

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

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Step 3.12.1

Multiply $\frac{u}{u^{2} - 3}$ by $\frac{u}{u}$ .

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

Step 3.12.2

Reorder the factors of $(u^{2} - 3) u$ .

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

Step 3.13

Combine the numerators over the common denominator.

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

Step 3.14

Multiply $u$ by $u$ by adding the exponents.

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Step 3.14.1

Move $u$ .

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

Step 3.14.2

Multiply $u$ by $u$ .

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

Step 3.15

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

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

Step 3.16

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

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Write the fraction using partial fraction decomposition.

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Step 4.3.2.1

Decompose the fraction and multiply through by the common denominator.

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Step 4.3.2.1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A}{u} + \frac{B u + C}{u^{2} - 3}$

Step 4.3.2.1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $u (u^{2} - 3)$ .

$\frac{(u^{2} + 1) (u (u^{2} - 3))}{u (u^{2} - 3)} = \frac{A (u (u^{2} - 3))}{u} + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.3

Cancel the common factor of $u$ .

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Step 4.3.2.1.3.1

Cancel the common factor.

$\frac{(u^{2} + 1) (u (u^{2} - 3))}{u (u^{2} - 3)} = \frac{A (u (u^{2} - 3))}{u} + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.3.2

Rewrite the expression.

$\frac{(u^{2} + 1) (u^{2} - 3)}{u^{2} - 3} = \frac{A (u (u^{2} - 3))}{u} + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.4

Cancel the common factor of $u^{2} - 3$ .

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Step 4.3.2.1.4.1

Cancel the common factor.

$\frac{(u^{2} + 1) (u^{2} - 3)}{u^{2} - 3} = \frac{A (u (u^{2} - 3))}{u} + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.4.2

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

$u^{2} + 1 = \frac{A (u (u^{2} - 3))}{u} + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.5

Simplify each term.

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Step 4.3.2.1.5.1

Cancel the common factor of $u$ .

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Step 4.3.2.1.5.1.1

Cancel the common factor.

$u^{2} + 1 = \frac{A (u (u^{2} - 3))}{u} + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.5.1.2

Divide $A (u^{2} - 3)$ by $1$ .

$u^{2} + 1 = A (u^{2} - 3) + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.5.2

Apply the distributive property.

$u^{2} + 1 = A u^{2} + A \cdot - 3 + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.5.3

Move $- 3$ to the left of $A$ .

$u^{2} + 1 = A u^{2} - 3 \cdot A + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.5.4

Cancel the common factor of $u^{2} - 3$ .

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Step 4.3.2.1.5.4.1

Cancel the common factor.

$u^{2} + 1 = A u^{2} - 3 A + \frac{(B u + C) (u (u^{2} - 3))}{u^{2} - 3}$

Step 4.3.2.1.5.4.2

Divide $(B u + C) (u)$ by $1$ .

$u^{2} + 1 = A u^{2} - 3 A + (B u + C) (u)$

Step 4.3.2.1.5.5

Apply the distributive property.

$u^{2} + 1 = A u^{2} - 3 A + B u \cdot u + C u$

Step 4.3.2.1.5.6

Multiply $u$ by $u$ by adding the exponents.

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Step 4.3.2.1.5.6.1

Move $u$ .

$u^{2} + 1 = A u^{2} - 3 A + B (u \cdot u) + C u$

Step 4.3.2.1.5.6.2

Multiply $u$ by $u$ .

$u^{2} + 1 = A u^{2} - 3 A + B u^{2} + C u$

Step 4.3.2.1.6

Move $- 3 A$ .

$u^{2} + 1 = A u^{2} + B u^{2} + C u - 3 A$

Step 4.3.2.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 4.3.2.2.1

Create an equation for the partial fraction variables by equating the coefficients of $u^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A + B$

Step 4.3.2.2.2

Create an equation for the partial fraction variables by equating the coefficients of $u$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = C$

Step 4.3.2.2.3

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $u$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = - 3 A$

Step 4.3.2.2.4

Set up the system of equations to find the coefficients of the partial fractions.

$1 = A + B$

$0 = C$

$1 = - 3 A$

$1 = A + B$

$0 = C$

$1 = - 3 A$

Step 4.3.2.3

Solve the system of equations.

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Step 4.3.2.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$1 = A + B$

$1 = - 3 A$

Step 4.3.2.3.2

Replace all occurrences of $C$ with $0$ in each equation.

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Step 4.3.2.3.2.1

Rewrite the equation as $- 3 A = 1$ .

$- 3 A = 1$

$C = 0$

$1 = A + B$

Step 4.3.2.3.2.2

Divide each term in $- 3 A = 1$ by $- 3$ and simplify.

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Step 4.3.2.3.2.2.1

Divide each term in $- 3 A = 1$ by $- 3$ .

$\frac{- 3 A}{- 3} = \frac{1}{- 3}$

$C = 0$

$1 = A + B$

Step 4.3.2.3.2.2.2

Simplify the left side.

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Step 4.3.2.3.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 A}{- 3} = \frac{1}{- 3}$

$C = 0$

$1 = A + B$

Step 4.3.2.3.2.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{- 3}$

$C = 0$

$1 = A + B$

$A = \frac{1}{- 3}$

$C = 0$

$1 = A + B$

$A = \frac{1}{- 3}$

$C = 0$

$1 = A + B$

Step 4.3.2.3.2.2.3

Simplify the right side.

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Step 4.3.2.3.2.2.3.1

Move the negative in front of the fraction.

$A = - \frac{1}{3}$

$C = 0$

$1 = A + B$

$A = - \frac{1}{3}$

$C = 0$

$1 = A + B$

$A = - \frac{1}{3}$

$C = 0$

$1 = A + B$

$A = - \frac{1}{3}$

$C = 0$

$1 = A + B$

Step 4.3.2.3.3

Replace all occurrences of $A$ with $- \frac{1}{3}$ in each equation.

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Step 4.3.2.3.3.1

Replace all occurrences of $A$ in $1 = A + B$ with $- \frac{1}{3}$ .

$1 = (- \frac{1}{3}) + B$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.3.2

Simplify the right side.

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Step 4.3.2.3.3.2.1

Remove parentheses.

$1 = - \frac{1}{3} + B$

$A = - \frac{1}{3}$

$C = 0$

$1 = - \frac{1}{3} + B$

$A = - \frac{1}{3}$

$C = 0$

$1 = - \frac{1}{3} + B$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.4

Solve for $B$ in $1 = - \frac{1}{3} + B$ .

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Step 4.3.2.3.4.1

Rewrite the equation as $- \frac{1}{3} + B = 1$ .

$- \frac{1}{3} + B = 1$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.4.2

Move all terms not containing $B$ to the right side of the equation.

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Step 4.3.2.3.4.2.1

Add $\frac{1}{3}$ to both sides of the equation.

$B = 1 + \frac{1}{3}$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.4.2.2

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

$B = \frac{3}{3} + \frac{1}{3}$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.4.2.3

Combine the numerators over the common denominator.

$B = \frac{3 + 1}{3}$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.4.2.4

Add $3$ and $1$ .

$B = \frac{4}{3}$

$A = - \frac{1}{3}$

$C = 0$

$B = \frac{4}{3}$

$A = - \frac{1}{3}$

$C = 0$

$B = \frac{4}{3}$

$A = - \frac{1}{3}$

$C = 0$

Step 4.3.2.3.5

Solve the system of equations.

$B = \frac{4}{3} A = - \frac{1}{3} C = 0$

Step 4.3.2.3.6

List all of the solutions.

$B = \frac{4}{3}, A = - \frac{1}{3}, C = 0$

Step 4.3.2.4

Replace each of the partial fraction coefficients in $\frac{A}{u} + \frac{B u + C}{u^{2} - 3}$ with the values found for $A$ , $B$ , and $C$ .

$- \frac{\frac{1}{3}}{u} + \frac{\frac{4}{3 u} + 0}{u^{2} - 3}$

Step 4.3.2.5

Simplify.

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Step 4.3.2.5.1

Remove parentheses.

$\frac{- \frac{1}{3}}{u} + \frac{(\frac{4}{3}) u + 0}{u^{2} - 3}$

Step 4.3.2.5.2

Simplify the numerator.

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Step 4.3.2.5.2.1

Combine $\frac{4}{3}$ and $u$ .

$\frac{- \frac{1}{3}}{u} + \frac{\frac{4 u}{3} + 0}{u^{2} - 3}$

Step 4.3.2.5.2.2

Add $\frac{4 u}{3}$ and $0$ .

$\frac{- \frac{1}{3}}{u} + \frac{\frac{4 u}{3}}{u^{2} - 3}$

Step 4.3.2.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{- \frac{1}{3}}{u} + \frac{4 u}{3} \cdot \frac{1}{u^{2} - 3}$

Step 4.3.2.5.4

Multiply $\frac{4 u}{3}$ by $\frac{1}{u^{2} - 3}$ .

$\frac{- \frac{1}{3}}{u} + \frac{4 u}{3 (u^{2} - 3)}$

Step 4.3.2.5.5

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{3} \cdot \frac{1}{u} + \frac{4 u}{3 (u^{2} - 3)}$

Step 4.3.2.5.6

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

$- \frac{1}{u \cdot 3} + \frac{4 u}{3 (u^{2} - 3)}$

Step 4.3.2.5.7

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

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

Step 4.3.3

Split the single integral into multiple integrals.

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

Since $\frac{4}{3}$ is constant with respect to $u$ , move $\frac{4}{3}$ out of the integral.

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

Step 4.3.8

Let $u_{1} = u^{2} - 3$ . Then $d u_{1} = 2 u d u$ , so $\frac{1}{2} d u_{1} = u d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.3.8.1

Let $u_{1} = u^{2} - 3$ . Find $\frac{d u_{1}}{d u}$ .

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Step 4.3.8.1.1

Differentiate $u^{2} - 3$ .

$\frac{d}{d u} [u^{2} - 3]$

Step 4.3.8.1.2

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

$\frac{d}{d u} [u^{2}] + \frac{d}{d u} [- 3]$

Step 4.3.8.1.3

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

$2 u + \frac{d}{d u} [- 3]$

Step 4.3.8.1.4

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

$2 u + 0$

Step 4.3.8.1.5

Add $2 u$ and $0$ .

$2 u$

Step 4.3.8.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 4.3.9

Simplify.

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Step 4.3.9.1

Multiply $\frac{1}{u_{1}}$ by $\frac{1}{2}$ .

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

Step 4.3.9.2

Move $2$ to the left of $u_{1}$ .

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

Step 4.3.10

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

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

Step 4.3.11

Simplify.

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Step 4.3.11.1

Multiply $\frac{1}{2}$ by $\frac{4}{3}$ .

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

Step 4.3.11.2

Multiply $2$ by $3$ .

$\ln (| y |) + C_{1} = - (- \frac{1}{3} (\ln (| u |) + C_{2}) + \frac{4}{6} \int \frac{1}{u_{1}} d u_{1})$

Step 4.3.11.3

Cancel the common factor of $4$ and $6$ .

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Step 4.3.11.3.1

Factor $2$ out of $4$ .

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

Step 4.3.11.3.2

Cancel the common factors.

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Step 4.3.11.3.2.1

Factor $2$ out of $6$ .

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

Step 4.3.11.3.2.2

Cancel the common factor.

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

Step 4.3.11.3.2.3

Rewrite the expression.

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

Step 4.3.12

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

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

Step 4.3.13

Simplify.

$\ln (| y |) + C_{1} = - (- \frac{1}{3} \ln (| u |) + \frac{2}{3} \ln (| u_{1} |)) + C_{4}$

Step 4.3.14

Replace all occurrences of $u_{1}$ with $u^{2} - 3$ .

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

Step 4.3.15

Simplify.

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Step 4.3.15.1

Simplify each term.

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Step 4.3.15.1.1

Combine $\ln (| u |)$ and $\frac{1}{3}$ .

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

Step 4.3.15.1.2

Combine $\frac{2}{3}$ and $\ln (| u^{2} - 3 |)$ .

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

Step 4.3.15.2

Combine the numerators over the common denominator.

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

Step 4.3.15.3

Factor $- 1$ out of $- \ln (| u |)$ .

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

Step 4.3.15.4

Factor $- 1$ out of $2 \ln (| u^{2} - 3 |)$ .

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

Step 4.3.15.5

Factor $- 1$ out of $- (\ln (| u |)) - (- 2 \ln (| u^{2} - 3 |))$ .

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

Step 4.3.15.6

Rewrite $- (\ln (| u |) - 2 \ln (| u^{2} - 3 |))$ as $- 1 (\ln (| u |) - 2 \ln (| u^{2} - 3 |))$ .

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

Step 4.3.15.7

Move the negative in front of the fraction.

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

Step 4.3.15.8

Multiply $- 1$ by $- 1$ .

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

Step 4.3.15.9

Multiply $\frac{\ln (| u |) - 2 \ln (| u^{2} - 3 |)}{3}$ by $1$ .

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

Step 4.3.16

Reorder terms.

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

Step 4.4

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

$\ln (| y |) = \frac{1}{3} (\ln (| u |) - 2 \ln (| u^{2} - 3 |)) + C$

Step 5

Solve for $y$ .

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Step 5.1

Simplify the right side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the distributive property.

$\ln (| y |) = \frac{1}{3} \ln (| u |) + \frac{1}{3} (- 2 \ln (| u^{2} - 3 |)) + C$

Step 5.1.1.2

Combine $\frac{1}{3}$ and $\ln (| u |)$ .

$\ln (| y |) = \frac{\ln (| u |)}{3} + \frac{1}{3} (- 2 \ln (| u^{2} - 3 |)) + C$

Step 5.1.1.3

Multiply $\frac{1}{3} (- 2 \ln (| u^{2} - 3 |))$ .

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Step 5.1.1.3.1

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 5.1.1.3.2

Combine $\frac{- 2}{3}$ and $\ln (| u^{2} - 3 |)$ .

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

Step 5.1.1.4

Move the negative in front of the fraction.

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

Step 5.2

Multiply each term in $\ln (| y |) = \frac{\ln (| u |)}{3} - \frac{2 \ln (| u^{2} - 3 |)}{3} + C$ by $3$ to eliminate the fractions.

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Step 5.2.1

Multiply each term in $\ln (| y |) = \frac{\ln (| u |)}{3} - \frac{2 \ln (| u^{2} - 3 |)}{3} + C$ by $3$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Move $3$ to the left of $\ln (| y |)$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $3$ .

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Step 5.2.3.1.1.1

Cancel the common factor.

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

Step 5.2.3.1.1.2

Rewrite the expression.

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

Step 5.2.3.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.1.2.1

Move the leading negative in $- \frac{2 \ln (| u^{2} - 3 |)}{3}$ into the numerator.

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.3

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

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

Step 5.3

Simplify the left side.

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Step 5.3.1

Simplify $3 \ln (| y |)$ by moving $3$ inside the logarithm.

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

Step 5.4

Simplify the right side.

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Step 5.4.1

Simplify $\ln (| u |) - 2 \ln (| u^{2} - 3 |) + 3 C$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

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

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

Step 5.4.1.1.2

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

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

Step 5.4.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 5.5

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

$\ln ({| y |}^{3}) - \ln (\frac{| u |}{{(u^{2} - 3)}^{2}}) = 3 C$

Step 5.6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{{| y |}^{3}}{\frac{| u |}{{(u^{2} - 3)}^{2}}}) = 3 C$

Step 5.7

Multiply the numerator by the reciprocal of the denominator.

$\ln ({| y |}^{3} (\frac{{(u^{2} - 3)}^{2}}{| u |})) = 3 C$

Step 5.8

Combine ${| y |}^{3}$ and $\frac{{(u^{2} - 3)}^{2}}{| u |}$ .

$\ln (\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |}) = 3 C$

Step 5.9

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

$e^{\ln (\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |})} = e^{3 C}$

Step 5.10

Rewrite $\ln (\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |}) = 3 C$ 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 $b^{y} = x$ .

$e^{3 C} = \frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |}$

Step 5.11

Solve for $y$ .

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Step 5.11.1

Rewrite the equation as $\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |} = e^{3 C}$ .

$\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |} = e^{3 C}$

Step 5.11.2

Multiply both sides by $| u |$ .

$\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |} | u | = e^{3 C} | u |$

Step 5.11.3

Simplify the left side.

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Step 5.11.3.1

Cancel the common factor of $| u |$ .

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Step 5.11.3.1.1

Cancel the common factor.

$\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{| u |} | u | = e^{3 C} | u |$

Step 5.11.3.1.2

Rewrite the expression.

${| y |}^{3} {(u^{2} - 3)}^{2} = e^{3 C} | u |$

Step 5.11.4

Solve for $y$ .

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Step 5.11.4.1

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

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Step 5.11.4.1.1

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

$\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{{(u^{2} - 3)}^{2}} = \frac{e^{3 C} | u |}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.1.2

Simplify the left side.

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Step 5.11.4.1.2.1

Cancel the common factor of ${(u^{2} - 3)}^{2}$ .

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Step 5.11.4.1.2.1.1

Cancel the common factor.

$\frac{{| y |}^{3} {(u^{2} - 3)}^{2}}{{(u^{2} - 3)}^{2}} = \frac{e^{3 C} | u |}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.1.2.1.2

Divide ${| y |}^{3}$ by $1$ .

${| y |}^{3} = \frac{e^{3 C} | u |}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.2

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

$| y | = \sqrt[3]{\frac{e^{3 C} | u |}{{(u^{2} - 3)}^{2}}}$

Step 5.11.4.3

Simplify $\sqrt[3]{\frac{e^{3 C} | u |}{{(u^{2} - 3)}^{2}}}$ .

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Step 5.11.4.3.1

Rewrite $\sqrt[3]{\frac{e^{3 C} | u |}{{(u^{2} - 3)}^{2}}}$ as $\frac{\sqrt[3]{e^{3 C} | u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}}$ .

$| y | = \frac{\sqrt[3]{e^{3 C} | u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}}$

Step 5.11.4.3.2

Simplify the numerator.

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Step 5.11.4.3.2.1

Rewrite $e^{3 C}$ as ${(e^{C})}^{3}$ .

$| y | = \frac{\sqrt[3]{{(e^{C})}^{3} | u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}}$

Step 5.11.4.3.2.2

Pull terms out from under the radical.

$| y | = \frac{e^{C} \sqrt[3]{| u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}}$

Step 5.11.4.3.3

Multiply $\frac{e^{C} \sqrt[3]{| u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}}$ by $\frac{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}} \cdot \frac{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}$

Step 5.11.4.3.4

Combine and simplify the denominator.

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Step 5.11.4.3.4.1

Multiply $\frac{e^{C} \sqrt[3]{| u |}}{\sqrt[3]{{(u^{2} - 3)}^{2}}}$ by $\frac{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{\sqrt[3]{{(u^{2} - 3)}^{2}} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}$

Step 5.11.4.3.4.2

Raise $\sqrt[3]{{(u^{2} - 3)}^{2}}$ to the power of $1$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{1} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}$

Step 5.11.4.3.4.3

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

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{1 + 2}}$

Step 5.11.4.3.4.4

Add $1$ and $2$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{\sqrt[3]{{(u^{2} - 3)}^{2}}}^{3}}$

Step 5.11.4.3.4.5

Rewrite ${\sqrt[3]{{(u^{2} - 3)}^{2}}}^{3}$ as ${(u^{2} - 3)}^{2}$ .

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Step 5.11.4.3.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(u^{2} - 3)}^{2}}$ as ${(u^{2} - 3)}^{\frac{2}{3}}$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{({(u^{2} - 3)}^{\frac{2}{3}})}^{3}}$

Step 5.11.4.3.4.5.2

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

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{2}{3} \cdot 3}}$

Step 5.11.4.3.4.5.3

Combine $\frac{2}{3}$ and $3$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{2 \cdot 3}{3}}}$

Step 5.11.4.3.4.5.4

Multiply $2$ by $3$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{6}{3}}}$

Step 5.11.4.3.4.5.5

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

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Step 5.11.4.3.4.5.5.1

Factor $3$ out of $6$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{3 \cdot 2}{3}}}$

Step 5.11.4.3.4.5.5.2

Cancel the common factors.

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Step 5.11.4.3.4.5.5.2.1

Factor $3$ out of $3$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{3 \cdot 2}{3 (1)}}}$

Step 5.11.4.3.4.5.5.2.2

Cancel the common factor.

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 5.11.4.3.4.5.5.2.3

Rewrite the expression.

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{\frac{2}{1}}}$

Step 5.11.4.3.4.5.5.2.4

Divide $2$ by $1$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} {\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5

Simplify the numerator.

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Step 5.11.4.3.5.1

Rewrite ${\sqrt[3]{{(u^{2} - 3)}^{2}}}^{2}$ as $\sqrt[3]{{({(u^{2} - 3)}^{2})}^{2}}$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{{({(u^{2} - 3)}^{2})}^{2}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.2

Multiply the exponents in ${({(u^{2} - 3)}^{2})}^{2}$ .

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Step 5.11.4.3.5.2.1

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

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{{(u^{2} - 3)}^{2 \cdot 2}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.2.2

Multiply $2$ by $2$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{{(u^{2} - 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.3

Use the Binomial Theorem.

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{{(u^{2})}^{4} + 4 {(u^{2})}^{3} \cdot - 3 + 6 {(u^{2})}^{2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4

Simplify each term.

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Step 5.11.4.3.5.4.1

Multiply the exponents in ${(u^{2})}^{4}$ .

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Step 5.11.4.3.5.4.1.1

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

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{2 \cdot 4} + 4 {(u^{2})}^{3} \cdot - 3 + 6 {(u^{2})}^{2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.1.2

Multiply $2$ by $4$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} + 4 {(u^{2})}^{3} \cdot - 3 + 6 {(u^{2})}^{2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.2

Multiply the exponents in ${(u^{2})}^{3}$ .

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Step 5.11.4.3.5.4.2.1

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

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} + 4 u^{2 \cdot 3} \cdot - 3 + 6 {(u^{2})}^{2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.2.2

Multiply $2$ by $3$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} + 4 u^{6} \cdot - 3 + 6 {(u^{2})}^{2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.3

Multiply $- 3$ by $4$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 6 {(u^{2})}^{2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.4

Multiply the exponents in ${(u^{2})}^{2}$ .

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Step 5.11.4.3.5.4.4.1

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

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 6 u^{2 \cdot 2} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.4.2

Multiply $2$ by $2$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 6 u^{4} {(- 3)}^{2} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.5

Raise $- 3$ to the power of $2$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 6 u^{4} \cdot 9 + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.6

Multiply $9$ by $6$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 54 u^{4} + 4 u^{2} {(- 3)}^{3} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.7

Raise $- 3$ to the power of $3$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 54 u^{4} + 4 u^{2} \cdot - 27 + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.8

Multiply $- 27$ by $4$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + {(- 3)}^{4}}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.4.9

Raise $- 3$ to the power of $4$ .

$| y | = \frac{e^{C} \sqrt[3]{| u |} \sqrt[3]{u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + 81}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.5.5

Combine using the product rule for radicals.

$| y | = \frac{e^{C} \sqrt[3]{(u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + 81) | u |}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.3.6

Reorder factors in $\frac{e^{C} \sqrt[3]{(u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + 81) | u |}}{{(u^{2} - 3)}^{2}}$ .

$| y | = \frac{e^{C} \sqrt[3]{| u | (u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + 81)}}{{(u^{2} - 3)}^{2}}$

Step 5.11.4.4

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

$y = \pm \frac{e^{C} \sqrt[3]{| u | (u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + 81)}}{{(u^{2} - 3)}^{2}}$

Step 6

Simplify the constant of integration.

$y = \pm \frac{C \sqrt[3]{| u | (u^{8} - 12 u^{6} + 54 u^{4} - 108 u^{2} + 81)}}{{(u^{2} - 3)}^{2}}$