Calculus Examples

$x (t^{2} - 1) d t + (t^{3} - 3 t) d x = 0$

Step 1

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

$(t^{3} - 3 t) d x = - x (t^{2} - 1) d t$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Factor $t^{3} - 3 t$ out of $(t^{3} - 3 t) x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

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

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

Step 3.3.2

Factor $x$ out of $(t^{3} - 3 t) x$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Factor $t$ out of $- 3 t$ .

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

Step 3.4.3

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

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

Step 3.5

Move the negative in front of the fraction.

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Cancel the common factor of $t$ .

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

Cancel the common factor.

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

Step 3.7.4

Rewrite the expression.

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

Step 3.8

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

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

Step 3.9

Multiply $- \frac{1}{t (t^{2} - 3)} \cdot - 1$ .

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Step 3.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.9.2

Multiply $\frac{1}{t (t^{2} - 3)}$ by $1$ .

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

Step 3.10

Move the negative in front of the fraction.

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

Step 3.11

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

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

Step 3.12

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

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Step 3.12.1

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

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

Step 3.12.2

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

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

Step 3.13

Combine the numerators over the common denominator.

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

Step 3.14

Simplify the numerator.

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Step 3.14.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.14.2

Rewrite $t \cdot t$ as $t^{2}$ .

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

Step 3.14.3

Reorder $- t^{2}$ and $1^{2}$ .

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

Step 3.14.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = t$ .

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Write the fraction using partial fraction decomposition.

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Step 4.3.1.1

Decompose the fraction and multiply through by the common denominator.

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Step 4.3.1.1.1

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.

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

Step 4.3.1.1.2

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

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

Step 4.3.1.1.3

Reduce the expression by cancelling the common factors.

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Step 4.3.1.1.3.1

Cancel the common factor of $t$ .

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Step 4.3.1.1.3.1.1

Cancel the common factor.

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

Step 4.3.1.1.3.1.2

Rewrite the expression.

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

Step 4.3.1.1.3.2

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

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Step 4.3.1.1.3.2.1

Cancel the common factor.

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

Step 4.3.1.1.3.2.2

Divide $(1 + t) (1 - t)$ by $1$ .

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

Step 4.3.1.1.4

Expand $(1 + t) (1 - t)$ using the FOIL Method.

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Step 4.3.1.1.4.1

Apply the distributive property.

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

Step 4.3.1.1.4.2

Apply the distributive property.

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

Step 4.3.1.1.4.3

Apply the distributive property.

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

Step 4.3.1.1.5

Simplify and combine like terms.

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Step 4.3.1.1.5.1

Simplify each term.

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Step 4.3.1.1.5.1.1

Multiply $1$ by $1$ .

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

Step 4.3.1.1.5.1.2

Multiply $- t$ by $1$ .

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

Step 4.3.1.1.5.1.3

Multiply $t$ by $1$ .

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

Step 4.3.1.1.5.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.1.5.1.5

Multiply $t$ by $t$ by adding the exponents.

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Step 4.3.1.1.5.1.5.1

Move $t$ .

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

Step 4.3.1.1.5.1.5.2

Multiply $t$ by $t$ .

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

Step 4.3.1.1.5.2

Add $- t$ and $t$ .

$1 + 0 - t^{2} = \frac{A (t (t^{2} - 3))}{t} + \frac{(B t + C) (t (t^{2} - 3))}{t^{2} - 3}$

Step 4.3.1.1.5.3

Add $1$ and $0$ .

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

Step 4.3.1.1.6

Reorder $1$ and $- t^{2}$ .

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

Step 4.3.1.1.7

Simplify each term.

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Step 4.3.1.1.7.1

Cancel the common factor of $t$ .

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Step 4.3.1.1.7.1.1

Cancel the common factor.

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

Step 4.3.1.1.7.1.2

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

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

Step 4.3.1.1.7.2

Apply the distributive property.

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

Step 4.3.1.1.7.3

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

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

Step 4.3.1.1.7.4

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

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Step 4.3.1.1.7.4.1

Cancel the common factor.

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

Step 4.3.1.1.7.4.2

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

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

Step 4.3.1.1.7.5

Apply the distributive property.

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

Step 4.3.1.1.7.6

Multiply $t$ by $t$ by adding the exponents.

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Step 4.3.1.1.7.6.1

Move $t$ .

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

Step 4.3.1.1.7.6.2

Multiply $t$ by $t$ .

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

Step 4.3.1.1.8

Move $- 3 A$ .

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $t^{2}$ 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 4.3.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of $t$ 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 4.3.1.2.3

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

$1 = - 3 A$

Step 4.3.1.2.4

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 4.3.1.3

Solve the system of equations.

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Step 4.3.1.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$- 1 = A + B$

$1 = - 3 A$

Step 4.3.1.3.2

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

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Step 4.3.1.3.2.1

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

$- 3 A = 1$

$C = 0$

$- 1 = A + B$

Step 4.3.1.3.2.2

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

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Step 4.3.1.3.2.2.1

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

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

$C = 0$

$- 1 = A + B$

Step 4.3.1.3.2.2.2

Simplify the left side.

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Step 4.3.1.3.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.1.3.2.2.2.1.1

Cancel the common factor.

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

$C = 0$

$- 1 = A + B$

Step 4.3.1.3.2.2.2.1.2

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 4.3.1.3.2.2.3

Simplify the right side.

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Step 4.3.1.3.2.2.3.1

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 4.3.1.3.3

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

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Step 4.3.1.3.3.1

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 4.3.1.3.3.2

Simplify the right side.

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Step 4.3.1.3.3.2.1

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 4.3.1.3.4

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

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Step 4.3.1.3.4.1

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

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

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

$C = 0$

Step 4.3.1.3.4.2

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

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Step 4.3.1.3.4.2.1

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

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

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

$C = 0$

Step 4.3.1.3.4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

$C = 0$

Step 4.3.1.3.4.2.3

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

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

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

$C = 0$

Step 4.3.1.3.4.2.4

Combine the numerators over the common denominator.

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

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

$C = 0$

Step 4.3.1.3.4.2.5

Simplify the numerator.

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Step 4.3.1.3.4.2.5.1

Multiply $- 1$ by $3$ .

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

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

$C = 0$

Step 4.3.1.3.4.2.5.2

Add $- 3$ and $1$ .

$B = \frac{- 2}{3}$

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

$C = 0$

$B = \frac{- 2}{3}$

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

$C = 0$

Step 4.3.1.3.4.2.6

Move the negative in front of the fraction.

$B = - \frac{2}{3}$

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

$C = 0$

$B = - \frac{2}{3}$

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

$C = 0$

$B = - \frac{2}{3}$

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

$C = 0$

Step 4.3.1.3.5

Solve the system of equations.

$B = - \frac{2}{3} A = - \frac{1}{3} C = 0$

Step 4.3.1.3.6

List all of the solutions.

$B = - \frac{2}{3}, A = - \frac{1}{3}, C = 0$

Step 4.3.1.4

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

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

Step 4.3.1.5

Simplify.

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Step 4.3.1.5.1

Remove parentheses.

$\frac{- \frac{1}{3}}{t} + \frac{(- \frac{2}{3}) \cdot t + 0}{t^{2} - 3}$

Step 4.3.1.5.2

Simplify the numerator.

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Step 4.3.1.5.2.1

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

$\frac{- \frac{1}{3}}{t} + \frac{- \frac{t \cdot 2}{3} + 0}{t^{2} - 3}$

Step 4.3.1.5.2.2

Move $2$ to the left of $t$ .

$\frac{- \frac{1}{3}}{t} + \frac{- \frac{2 \cdot t}{3} + 0}{t^{2} - 3}$

Step 4.3.1.5.2.3

Add $- \frac{2 t}{3}$ and $0$ .

$\frac{- \frac{1}{3}}{t} + \frac{- \frac{2 t}{3}}{t^{2} - 3}$

Step 4.3.1.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{- \frac{1}{3}}{t} - \frac{2 t}{3} \cdot \frac{1}{t^{2} - 3}$

Step 4.3.1.5.4

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

$\frac{- \frac{1}{3}}{t} - \frac{2 t}{(t^{2} - 3) \cdot 3}$

Step 4.3.1.5.5

Move $3$ to the left of $t^{2} - 3$ .

$\frac{- \frac{1}{3}}{t} - \frac{2 t}{3 (t^{2} - 3)}$

Step 4.3.1.5.6

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{3} \cdot \frac{1}{t} - \frac{2 t}{3 (t^{2} - 3)}$

Step 4.3.1.5.7

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

$- \frac{1}{t \cdot 3} - \frac{2 t}{3 (t^{2} - 3)}$

Step 4.3.1.5.8

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

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

Step 4.3.2

Split the single integral into multiple integrals.

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

Since $\frac{2}{3}$ is constant with respect to $t$ , move $\frac{2}{3}$ out of the integral.

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

Step 4.3.8

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

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Step 4.3.8.1

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

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Step 4.3.8.1.1

Differentiate $t^{2} - 3$ .

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

Step 4.3.8.1.2

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

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

Step 4.3.8.1.3

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

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

Step 4.3.8.1.4

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

$2 t + 0$

Step 4.3.8.1.5

Add $2 t$ and $0$ .

$2 t$

Step 4.3.8.2

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

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

Step 4.3.9

Simplify.

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Step 4.3.9.1

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

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

Step 4.3.9.2

Move $2$ to the left of $u$ .

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

Step 4.3.10

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

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

Step 4.3.11

Simplify.

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Step 4.3.11.1

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

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

Step 4.3.11.2

Multiply $2$ by $3$ .

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

Step 4.3.11.3

Cancel the common factor of $2$ and $6$ .

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Step 4.3.11.3.1

Factor $2$ out of $2$ .

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

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 $6$ .

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

Step 4.3.11.3.2.2

Cancel the common factor.

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

Step 4.3.11.3.2.3

Rewrite the expression.

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

Step 4.3.12

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

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

Step 4.3.13

Simplify.

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

Step 4.3.14

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

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

Step 4.4

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

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

Step 5

Solve for $x$ .

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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

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

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

Step 5.1.1.2

Combine $\ln (| t^{2} - 3 |)$ and $\frac{1}{3}$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

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

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

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

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

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

Step 5.2.3.1.1.2

Cancel the common factor.

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

Step 5.2.3.1.1.3

Rewrite the expression.

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

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 $- \frac{\ln (| t^{2} - 3 |)}{3}$ into the numerator.

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.3

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

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

Step 5.3

Simplify the left side.

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Step 5.3.1

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 5.8

Apply the distributive property.

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

Step 5.9

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

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Step 5.9.1

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

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Step 5.9.1.1

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

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

Step 5.9.1.2

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

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

Step 5.9.2

Add $2$ and $1$ .

$\ln ({| x |}^{3} | t^{3} - 3 t |) = 3 C$

Step 5.10

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

$e^{\ln ({| x |}^{3} | t^{3} - 3 t |)} = e^{3 C}$

Step 5.11

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

$e^{3 C} = {| x |}^{3} | t^{3} - 3 t |$

Step 5.12

Solve for $x$ .

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Step 5.12.1

Rewrite the equation as ${| x |}^{3} | t^{3} - 3 t | = e^{3 C}$ .

${| x |}^{3} | t^{3} - 3 t | = e^{3 C}$

Step 5.12.2

Divide each term in ${| x |}^{3} | t^{3} - 3 t | = e^{3 C}$ by $| t^{3} - 3 t |$ and simplify.

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Step 5.12.2.1

Divide each term in ${| x |}^{3} | t^{3} - 3 t | = e^{3 C}$ by $| t^{3} - 3 t |$ .

$\frac{{| x |}^{3} | t^{3} - 3 t |}{| t^{3} - 3 t |} = \frac{e^{3 C}}{| t^{3} - 3 t |}$

Step 5.12.2.2

Simplify the left side.

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Step 5.12.2.2.1

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

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Step 5.12.2.2.1.1

Cancel the common factor.

$\frac{{| x |}^{3} | t^{3} - 3 t |}{| t^{3} - 3 t |} = \frac{e^{3 C}}{| t^{3} - 3 t |}$

Step 5.12.2.2.1.2

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

${| x |}^{3} = \frac{e^{3 C}}{| t^{3} - 3 t |}$

Step 5.12.2.3

Simplify the right side.

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Step 5.12.2.3.1

Simplify the denominator.

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Step 5.12.2.3.1.1

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

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Step 5.12.2.3.1.1.1

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

${| x |}^{3} = \frac{e^{3 C}}{| t \cdot t^{2} - 3 t |}$

Step 5.12.2.3.1.1.2

Factor $t$ out of $- 3 t$ .

${| x |}^{3} = \frac{e^{3 C}}{| t \cdot t^{2} + t \cdot - 3 |}$

Step 5.12.2.3.1.1.3

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

${| x |}^{3} = \frac{e^{3 C}}{| t (t^{2} - 3) |}$

Step 5.12.2.3.1.2

Apply the distributive property.

${| x |}^{3} = \frac{e^{3 C}}{| t \cdot t^{2} + t \cdot - 3 |}$

Step 5.12.2.3.1.3

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

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Step 5.12.2.3.1.3.1

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

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Step 5.12.2.3.1.3.1.1

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

${| x |}^{3} = \frac{e^{3 C}}{| t^{1} t^{2} + t \cdot - 3 |}$

Step 5.12.2.3.1.3.1.2

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

${| x |}^{3} = \frac{e^{3 C}}{| t^{1 + 2} + t \cdot - 3 |}$

Step 5.12.2.3.1.3.2

Add $1$ and $2$ .

${| x |}^{3} = \frac{e^{3 C}}{| t^{3} + t \cdot - 3 |}$

Step 5.12.2.3.1.4

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

${| x |}^{3} = \frac{e^{3 C}}{| t^{3} - 3 \cdot t |}$

Step 5.12.2.3.1.5

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

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Step 5.12.2.3.1.5.1

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

${| x |}^{3} = \frac{e^{3 C}}{| t \cdot t^{2} - 3 t |}$

Step 5.12.2.3.1.5.2

Factor $t$ out of $- 3 t$ .

${| x |}^{3} = \frac{e^{3 C}}{| t \cdot t^{2} + t \cdot - 3 |}$

Step 5.12.2.3.1.5.3

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

${| x |}^{3} = \frac{e^{3 C}}{| t (t^{2} - 3) |}$

Step 5.12.3

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

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

Step 5.12.4

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

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Step 5.12.4.1

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

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

Step 5.12.4.2

Simplify the numerator.

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Step 5.12.4.2.1

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

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

Step 5.12.4.2.2

Pull terms out from under the radical, assuming real numbers.

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

Step 5.12.4.3

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

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

Step 5.12.4.4

Combine and simplify the denominator.

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Step 5.12.4.4.1

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

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

Step 5.12.4.4.2

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

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

Step 5.12.4.4.3

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

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

Step 5.12.4.4.4

Add $1$ and $2$ .

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

Step 5.12.4.4.5

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

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Step 5.12.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{| t (t^{2} - 3) |}$ as ${| t (t^{2} - 3) |}^{\frac{1}{3}}$ .

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

Step 5.12.4.4.5.2

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

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

Step 5.12.4.4.5.3

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

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

Step 5.12.4.4.5.4

Cancel the common factor of $3$ .

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Step 5.12.4.4.5.4.1

Cancel the common factor.

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

Step 5.12.4.4.5.4.2

Rewrite the expression.

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

Step 5.12.4.4.5.5

Simplify.

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

Step 5.12.4.5

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

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

Step 5.12.5

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

$x = \pm \frac{e^{C} \sqrt[3]{{| t (t^{2} - 3) |}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6

Simplify the numerator.

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Step 5.12.6.1

Apply the distributive property.

$x = \pm \frac{e^{C} \sqrt[3]{{| t \cdot t^{2} + t \cdot - 3 |}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.2

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

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Step 5.12.6.2.1

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

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Step 5.12.6.2.1.1

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

$x = \pm \frac{e^{C} \sqrt[3]{{| t^{1} t^{2} + t \cdot - 3 |}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.2.1.2

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

$x = \pm \frac{e^{C} \sqrt[3]{{| t^{1 + 2} + t \cdot - 3 |}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.2.2

Add $1$ and $2$ .

$x = \pm \frac{e^{C} \sqrt[3]{{| t^{3} + t \cdot - 3 |}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.3

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

$x = \pm \frac{e^{C} \sqrt[3]{{| t^{3} - 3 \cdot t |}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.4

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

$x = \pm \frac{e^{C} \sqrt[3]{{(t^{3} - 3 t)}^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.5

Rewrite ${(t^{3} - 3 t)}^{2}$ as $(t^{3} - 3 t) (t^{3} - 3 t)$ .

$x = \pm \frac{e^{C} \sqrt[3]{(t^{3} - 3 t) (t^{3} - 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.6

Expand $(t^{3} - 3 t) (t^{3} - 3 t)$ using the FOIL Method.

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Step 5.12.6.6.1

Apply the distributive property.

$x = \pm \frac{e^{C} \sqrt[3]{t^{3} (t^{3} - 3 t) - 3 t (t^{3} - 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.6.2

Apply the distributive property.

$x = \pm \frac{e^{C} \sqrt[3]{t^{3} t^{3} + t^{3} (- 3 t) - 3 t (t^{3} - 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.6.3

Apply the distributive property.

$x = \pm \frac{e^{C} \sqrt[3]{t^{3} t^{3} + t^{3} (- 3 t) - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7

Simplify and combine like terms.

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Step 5.12.6.7.1

Simplify each term.

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Step 5.12.6.7.1.1

Multiply $t^{3}$ by $t^{3}$ by adding the exponents.

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Step 5.12.6.7.1.1.1

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

$x = \pm \frac{e^{C} \sqrt[3]{t^{3 + 3} + t^{3} (- 3 t) - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.1.2

Add $3$ and $3$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} + t^{3} (- 3 t) - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.2

Rewrite using the commutative property of multiplication.

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{3} t - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.3

Multiply $t^{3}$ by $t$ by adding the exponents.

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Step 5.12.6.7.1.3.1

Move $t$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 (t \cdot t^{3}) - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.3.2

Multiply $t$ by $t^{3}$ .

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Step 5.12.6.7.1.3.2.1

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

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 (t^{1} t^{3}) - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.3.2.2

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

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{1 + 3} - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.3.3

Add $1$ and $3$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t \cdot t^{3} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.4

Multiply $t$ by $t^{3}$ by adding the exponents.

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Step 5.12.6.7.1.4.1

Move $t^{3}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 (t^{3} t) - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.4.2

Multiply $t^{3}$ by $t$ .

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Step 5.12.6.7.1.4.2.1

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

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 (t^{3} t^{1}) - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.4.2.2

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

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t^{3 + 1} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.4.3

Add $3$ and $1$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t^{4} - 3 t (- 3 t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.5

Rewrite using the commutative property of multiplication.

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t^{4} - 3 \cdot - 3 t \cdot t}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.6

Multiply $t$ by $t$ by adding the exponents.

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Step 5.12.6.7.1.6.1

Move $t$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t^{4} - 3 \cdot - 3 (t \cdot t)}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.6.2

Multiply $t$ by $t$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t^{4} - 3 \cdot - 3 t^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.1.7

Multiply $- 3$ by $- 3$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 3 t^{4} - 3 t^{4} + 9 t^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.7.2

Subtract $3 t^{4}$ from $- 3 t^{4}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{6} - 6 t^{4} + 9 t^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.8

Factor $t^{2}$ out of $t^{6} - 6 t^{4} + 9 t^{2}$ .

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Step 5.12.6.8.1

Factor $t^{2}$ out of $t^{6}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} t^{4} - 6 t^{4} + 9 t^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.8.2

Factor $t^{2}$ out of $- 6 t^{4}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} t^{4} + t^{2} (- 6 t^{2}) + 9 t^{2}}}{| t (t^{2} - 3) |}$

Step 5.12.6.8.3

Factor $t^{2}$ out of $9 t^{2}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} t^{4} + t^{2} (- 6 t^{2}) + t^{2} \cdot 9}}{| t (t^{2} - 3) |}$

Step 5.12.6.8.4

Factor $t^{2}$ out of $t^{2} t^{4} + t^{2} (- 6 t^{2})$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} (t^{4} - 6 t^{2}) + t^{2} \cdot 9}}{| t (t^{2} - 3) |}$

Step 5.12.6.8.5

Factor $t^{2}$ out of $t^{2} (t^{4} - 6 t^{2}) + t^{2} \cdot 9$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} (t^{4} - 6 t^{2} + 9)}}{| t (t^{2} - 3) |}$

Step 5.12.6.9

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} ({(t^{2})}^{2} - 6 t^{2} + 9)}}{| t (t^{2} - 3) |}$

Step 5.12.6.10

Let $u = t^{2}$ . Substitute $u$ for all occurrences of $t^{2}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} (u^{2} - 6 u + 9)}}{| t (t^{2} - 3) |}$

Step 5.12.6.11

Factor using the perfect square rule.

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Step 5.12.6.11.1

Rewrite $9$ as $3^{2}$ .

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

Step 5.12.6.11.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 u = 2 \cdot u \cdot 3$

Step 5.12.6.11.3

Rewrite the polynomial.

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

Step 5.12.6.11.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 3$ .

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

Step 5.12.6.12

Replace all occurrences of $u$ with $t^{2}$ .

$x = \pm \frac{e^{C} \sqrt[3]{t^{2} {(t^{2} - 3)}^{2}}}{| t (t^{2} - 3) |}$

Step 6

Simplify the constant of integration.

$x = \pm \frac{C \sqrt[3]{t^{2} {(t^{2} - 3)}^{2}}}{| t (t^{2} - 3) |}$