Calculus Examples

$\int \frac{2 x^{3} - 2}{(x^{4} - 4 x)} d x$

Step 1

Write the fraction using partial fraction decomposition.

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Step 1.1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1.1

Factor the fraction.

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Step 1.1.1.1

Factor $2$ out of $2 x^{3} - 2$ .

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Step 1.1.1.1.1

Factor $2$ out of $2 x^{3}$ .

$\frac{2 (x^{3}) - 2}{x^{4} - 4 x}$

Step 1.1.1.1.2

Factor $2$ out of $- 2$ .

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

Step 1.1.1.1.3

Factor $2$ out of $2 (x^{3}) + 2 (- 1)$ .

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

Step 1.1.1.2

Rewrite $1$ as $1^{3}$ .

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

Step 1.1.1.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 1.1.1.4

Factor.

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Step 1.1.1.4.1

Simplify.

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Step 1.1.1.4.1.1

Multiply $x$ by $1$ .

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

Step 1.1.1.4.1.2

One to any power is one.

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

Step 1.1.1.4.2

Remove unnecessary parentheses.

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

Step 1.1.1.5

Factor $x$ out of $x^{4} - 4 x$ .

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Step 1.1.1.5.1

Factor $x$ out of $x^{4}$ .

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

Step 1.1.1.5.2

Factor $x$ out of $- 4 x$ .

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

Step 1.1.1.5.3

Factor $x$ out of $x \cdot x^{3} + x \cdot - 4$ .

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

Step 1.1.2

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 3rd order, $3$ 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.

$\frac{A}{x} + \frac{B x^{2} + C x + D}{x^{3} - 4}$

Step 1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x (x^{3} - 4)$ .

$\frac{2 (x - 1) (x^{2} + x + 1) (x (x^{3} - 4))}{x (x^{3} - 4)} = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.4

Simplify terms.

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Step 1.1.4.1

Cancel the common factor of $x$ .

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Step 1.1.4.1.1

Cancel the common factor.

$\frac{2 (x - 1) (x^{2} + x + 1) (x (x^{3} - 4))}{x (x^{3} - 4)} = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.4.1.2

Rewrite the expression.

$\frac{2 (x - 1) (x^{2} + x + 1) (x^{3} - 4)}{x^{3} - 4} = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.4.2

Cancel the common factor of $x^{3} - 4$ .

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Step 1.1.4.2.1

Cancel the common factor.

$\frac{2 (x - 1) (x^{2} + x + 1) (x^{3} - 4)}{x^{3} - 4} = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.4.2.2

Divide $2 (x - 1) (x^{2} + x + 1)$ by $1$ .

$2 (x - 1) (x^{2} + x + 1) = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.4.3

Apply the distributive property.

$(2 x + 2 \cdot - 1) (x^{2} + x + 1) = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.4.4

Multiply $2$ by $- 1$ .

$(2 x - 2) (x^{2} + x + 1) = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.5

Expand $(2 x - 2) (x^{2} + x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 x \cdot x^{2} + 2 x \cdot x + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6

Simplify terms.

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Step 1.1.6.1

Simplify each term.

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Step 1.1.6.1.1

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

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Step 1.1.6.1.1.1

Move $x^{2}$ .

$2 (x^{2} x) + 2 x \cdot x + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.1.2

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

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Step 1.1.6.1.1.2.1

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

$2 (x^{2} x) + 2 x \cdot x + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.1.2.2

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

$2 x^{2 + 1} + 2 x \cdot x + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.1.3

Add $2$ and $1$ .

$2 x^{3} + 2 x \cdot x + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.2

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

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Step 1.1.6.1.2.1

Move $x$ .

$2 x^{3} + 2 (x \cdot x) + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.2.2

Multiply $x$ by $x$ .

$2 x^{3} + 2 x^{2} + 2 x \cdot 1 - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.3

Multiply $2$ by $1$ .

$2 x^{3} + 2 x^{2} + 2 x - 2 x^{2} - 2 x - 2 \cdot 1 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.1.4

Multiply $- 2$ by $1$ .

$2 x^{3} + 2 x^{2} + 2 x - 2 x^{2} - 2 x - 2 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.2

Combine the opposite terms in $2 x^{3} + 2 x^{2} + 2 x - 2 x^{2} - 2 x - 2$ .

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Step 1.1.6.2.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$2 x^{3} + 2 x + 0 - 2 x - 2 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.2.2

Add $2 x^{3} + 2 x$ and $0$ .

$2 x^{3} + 2 x - 2 x - 2 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.2.3

Subtract $2 x$ from $2 x$ .

$2 x^{3} + 0 - 2 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.6.2.4

Add $2 x^{3}$ and $0$ .

$2 x^{3} - 2 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.7

Simplify each term.

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Step 1.1.7.1

Cancel the common factor of $x$ .

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Step 1.1.7.1.1

Cancel the common factor.

$2 x^{3} - 2 = \frac{A (x (x^{3} - 4))}{x} + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.7.1.2

Divide $A (x^{3} - 4)$ by $1$ .

$2 x^{3} - 2 = A (x^{3} - 4) + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.7.2

Apply the distributive property.

$2 x^{3} - 2 = A x^{3} + A \cdot - 4 + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.7.3

Move $- 4$ to the left of $A$ .

$2 x^{3} - 2 = A x^{3} - 4 \cdot A + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.7.4

Cancel the common factor of $x^{3} - 4$ .

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Step 1.1.7.4.1

Cancel the common factor.

$2 x^{3} - 2 = A x^{3} - 4 A + \frac{(B x^{2} + C x + D) (x (x^{3} - 4))}{x^{3} - 4}$

Step 1.1.7.4.2

Divide $(B x^{2} + C x + D) (x)$ by $1$ .

$2 x^{3} - 2 = A x^{3} - 4 A + (B x^{2} + C x + D) (x)$

Step 1.1.7.5

Apply the distributive property.

$2 x^{3} - 2 = A x^{3} - 4 A + B x^{2} x + C x \cdot x + D x$

Step 1.1.7.6

Simplify.

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Step 1.1.7.6.1

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

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Step 1.1.7.6.1.1

Move $x$ .

$2 x^{3} - 2 = A x^{3} - 4 A + B (x \cdot x^{2}) + C x \cdot x + D x$

Step 1.1.7.6.1.2

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

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Step 1.1.7.6.1.2.1

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

$2 x^{3} - 2 = A x^{3} - 4 A + B (x \cdot x^{2}) + C x \cdot x + D x$

Step 1.1.7.6.1.2.2

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

$2 x^{3} - 2 = A x^{3} - 4 A + B x^{1 + 2} + C x \cdot x + D x$

Step 1.1.7.6.1.3

Add $1$ and $2$ .

$2 x^{3} - 2 = A x^{3} - 4 A + B x^{3} + C x \cdot x + D x$

Step 1.1.7.6.2

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

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Step 1.1.7.6.2.1

Move $x$ .

$2 x^{3} - 2 = A x^{3} - 4 A + B x^{3} + C (x \cdot x) + D x$

Step 1.1.7.6.2.2

Multiply $x$ by $x$ .

$2 x^{3} - 2 = A x^{3} - 4 A + B x^{3} + C x^{2} + D x$

Step 1.1.8

Move $- 4 A$ .

$2 x^{3} - 2 = A x^{3} + B x^{3} + C x^{2} + D x - 4 A$

Step 1.2

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

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Step 1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$2 = A + B$

Step 1.2.2

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

$0 = C$

Step 1.2.3

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

$0 = D$

Step 1.2.4

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

$- 2 = - 4 A$

Step 1.2.5

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

$2 = A + B$

$0 = C$

$0 = D$

$- 2 = - 4 A$

$2 = A + B$

$0 = C$

$0 = D$

$- 2 = - 4 A$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$2 = A + B$

$0 = D$

$- 2 = - 4 A$

Step 1.3.2

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

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Step 1.3.2.1

Rewrite the equation as $D = 0$ .

$D = 0$

$C = 0$

$2 = A + B$

$- 2 = - 4 A$

Step 1.3.2.2

Rewrite the equation as $- 4 A = - 2$ .

$- 4 A = - 2$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3

Divide each term in $- 4 A = - 2$ by $- 4$ and simplify.

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Step 1.3.2.3.1

Divide each term in $- 4 A = - 2$ by $- 4$ .

$\frac{- 4 A}{- 4} = \frac{- 2}{- 4}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3.2

Simplify the left side.

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Step 1.3.2.3.2.1

Cancel the common factor of $- 4$ .

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Step 1.3.2.3.2.1.1

Cancel the common factor.

$\frac{- 4 A}{- 4} = \frac{- 2}{- 4}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3.2.1.2

Divide $A$ by $1$ .

$A = \frac{- 2}{- 4}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{- 2}{- 4}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{- 2}{- 4}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3.3

Simplify the right side.

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Step 1.3.2.3.3.1

Cancel the common factor of $- 2$ and $- 4$ .

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Step 1.3.2.3.3.1.1

Factor $- 2$ out of $- 2$ .

$A = \frac{- 2 \cdot 1}{- 4}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3.3.1.2

Cancel the common factors.

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Step 1.3.2.3.3.1.2.1

Factor $- 2$ out of $- 4$ .

$A = \frac{- 2 \cdot 1}{- 2 \cdot 2}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3.3.1.2.2

Cancel the common factor.

$A = \frac{- 2 \cdot 1}{- 2 \cdot 2}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.2.3.3.1.2.3

Rewrite the expression.

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = A + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = A + B$

Step 1.3.3

Replace all occurrences of $A$ with $\frac{1}{2}$ in each equation.

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Step 1.3.3.1

Replace all occurrences of $A$ in $2 = A + B$ with $\frac{1}{2}$ .

$2 = (\frac{1}{2}) + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.3.2

Simplify the right side.

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Step 1.3.3.2.1

Remove parentheses.

$2 = \frac{1}{2} + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = \frac{1}{2} + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

$2 = \frac{1}{2} + B$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4

Solve for $B$ in $2 = \frac{1}{2} + B$ .

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Step 1.3.4.1

Rewrite the equation as $\frac{1}{2} + B = 2$ .

$\frac{1}{2} + B = 2$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4.2

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

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Step 1.3.4.2.1

Subtract $\frac{1}{2}$ from both sides of the equation.

$B = 2 - \frac{1}{2}$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4.2.2

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

$B = 2 \cdot \frac{2}{2} - \frac{1}{2}$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4.2.3

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

$B = \frac{2 \cdot 2}{2} - \frac{1}{2}$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4.2.4

Combine the numerators over the common denominator.

$B = \frac{2 \cdot 2 - 1}{2}$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4.2.5

Simplify the numerator.

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Step 1.3.4.2.5.1

Multiply $2$ by $2$ .

$B = \frac{4 - 1}{2}$

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.4.2.5.2

Subtract $1$ from $4$ .

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

$A = \frac{1}{2}$

$D = 0$

$C = 0$

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

$A = \frac{1}{2}$

$D = 0$

$C = 0$

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

$A = \frac{1}{2}$

$D = 0$

$C = 0$

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

$A = \frac{1}{2}$

$D = 0$

$C = 0$

Step 1.3.5

Solve the system of equations.

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

Step 1.3.6

List all of the solutions.

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

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A}{x} + \frac{B x^{2} + C x + D}{x^{3} - 4}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Add $\frac{3}{2} x^{2} + 0 \cdot x$ and $0$ .

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

Step 1.5.1.2

Factor $x$ out of $\frac{3}{2} x^{2} + 0 \cdot x$ .

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Step 1.5.1.2.1

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

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

Step 1.5.1.2.2

Factor $x$ out of $0 \cdot x$ .

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

Step 1.5.1.2.3

Factor $x$ out of $x (\frac{3}{2} x) + x \cdot 0$ .

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

Step 1.5.1.3

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

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

Step 1.5.1.4

Add $\frac{3 x}{2}$ and $0$ .

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

Step 1.5.2

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

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

Step 1.5.3

Simplify the numerator.

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Step 1.5.3.1

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

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

Step 1.5.3.2

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

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

Step 1.5.3.3

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

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

Step 1.5.3.4

Add $1$ and $1$ .

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

Step 1.5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.5

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

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

Step 1.5.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.7

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

$\int \frac{1}{2 x} + \frac{3 x^{2}}{2 (x^{3} - 4)} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{1}{2 x} d x + \int \frac{3 x^{2}}{2 (x^{3} - 4)} d x$

Step 3

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

$\frac{1}{2} \int \frac{1}{x} d x + \int \frac{3 x^{2}}{2 (x^{3} - 4)} d x$

Step 4

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

$\frac{1}{2} (\ln (| x |) + C) + \int \frac{3 x^{2}}{2 (x^{3} - 4)} d x$

Step 5

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{2} \int \frac{x^{2}}{x^{3} - 4} d x$

Step 6

Let $u = x^{3} - 4$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = x^{3} - 4$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $x^{3} - 4$ .

$\frac{d}{d x} [x^{3} - 4]$

Step 6.1.2

By the Sum Rule, the derivative of $x^{3} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 4]$

Step 6.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 6.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 6.2

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{2} \int \frac{1}{u} \cdot \frac{1}{3} d u$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{2} \int \frac{1}{u \cdot 3} d u$

Step 7.2

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{2} \int \frac{1}{3 u} d u$

Step 8

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{2} (\frac{1}{3} \int \frac{1}{u} d u)$

Step 9

Simplify.

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Step 9.1

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{3 \cdot 2} \int \frac{1}{u} d u$

Step 9.2

Multiply $3$ by $2$ .

$\frac{1}{2} (\ln (| x |) + C) + \frac{3}{6} \int \frac{1}{u} d u$

Step 9.3

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

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Step 9.3.1

Factor $3$ out of $3$ .

$\frac{1}{2} (\ln (| x |) + C) + \frac{3 (1)}{6} \int \frac{1}{u} d u$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $3$ out of $6$ .

$\frac{1}{2} (\ln (| x |) + C) + \frac{3 \cdot 1}{3 \cdot 2} \int \frac{1}{u} d u$

Step 9.3.2.2

Cancel the common factor.

$\frac{1}{2} (\ln (| x |) + C) + \frac{3 \cdot 1}{3 \cdot 2} \int \frac{1}{u} d u$

Step 9.3.2.3

Rewrite the expression.

$\frac{1}{2} (\ln (| x |) + C) + \frac{1}{2} \int \frac{1}{u} d u$

Step 10

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

$\frac{1}{2} (\ln (| x |) + C) + \frac{1}{2} (\ln (| u |) + C)$

Step 11

Simplify.

$\frac{1}{2} \ln (| x |) + \frac{1}{2} \ln (| u |) + C$

Step 12

Replace all occurrences of $u$ with $x^{3} - 4$ .

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