Calculus Examples

$\int \frac{x^{2} + 18}{x^{3} - 9 x} d x$

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 $x$ out of $x^{3} - 9 x$ .

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Step 1.1.1.1.1

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

$\frac{x^{2} + 18}{x \cdot x^{2} - 9 x}$

Step 1.1.1.1.2

Factor $x$ out of $- 9 x$ .

$\frac{x^{2} + 18}{x \cdot x^{2} + x \cdot - 9}$

Step 1.1.1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

$\frac{x^{2} + 18}{x (x^{2} - 9)}$

Step 1.1.1.2

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

$\frac{x^{2} + 18}{x (x^{2} - 3^{2})}$

Step 1.1.1.3

Factor.

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Step 1.1.1.3.1

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

$\frac{x^{2} + 18}{x ((x + 3) (x - 3))}$

Step 1.1.1.3.2

Remove unnecessary parentheses.

$\frac{x^{2} + 18}{x (x + 3) (x - 3)}$

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 in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{x} + \frac{B}{x + 3}$

Step 1.1.3

$\frac{A}{x} + \frac{B}{x + 3} + \frac{C}{x - 3}$

Step 1.1.4

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

$\frac{(x^{2} + 18) (x (x + 3) (x - 3))}{x (x + 3) (x - 3)} = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.5

Cancel the common factor of $x$ .

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Step 1.1.5.1

Cancel the common factor.

$\frac{(x^{2} + 18) (x (x + 3) (x - 3))}{x (x + 3) (x - 3)} = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.5.2

Rewrite the expression.

$\frac{(x^{2} + 18) ((x + 3) (x - 3))}{(x + 3) (x - 3)} = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.6

Cancel the common factor of $x + 3$ .

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Step 1.1.6.1

Cancel the common factor.

$\frac{(x^{2} + 18) ((x + 3) (x - 3))}{(x + 3) (x - 3)} = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.6.2

Rewrite the expression.

$\frac{(x^{2} + 18) (x - 3)}{x - 3} = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.7

Cancel the common factor of $x - 3$ .

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Step 1.1.7.1

Cancel the common factor.

$\frac{(x^{2} + 18) (x - 3)}{x - 3} = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.7.2

Divide $x^{2} + 18$ by $1$ .

$x^{2} + 18 = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8

Simplify each term.

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Step 1.1.8.1

Cancel the common factor of $x$ .

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Step 1.1.8.1.1

Cancel the common factor.

$x^{2} + 18 = \frac{A (x (x + 3) (x - 3))}{x} + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.1.2

Divide $A ((x + 3) (x - 3))$ by $1$ .

$x^{2} + 18 = A ((x + 3) (x - 3)) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.2

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 1.1.8.2.1

Apply the distributive property.

$x^{2} + 18 = A (x (x - 3) + 3 (x - 3)) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.2.2

Apply the distributive property.

$x^{2} + 18 = A (x \cdot x + x \cdot - 3 + 3 (x - 3)) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.2.3

Apply the distributive property.

$x^{2} + 18 = A (x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.3

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 1.1.8.3.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$x^{2} + 18 = A (x \cdot x - 3 x + 3 x + 3 \cdot - 3) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.3.2

Add $- 3 x$ and $3 x$ .

$x^{2} + 18 = A (x \cdot x + 0 + 3 \cdot - 3) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.3.3

Add $x \cdot x$ and $0$ .

$x^{2} + 18 = A (x \cdot x + 3 \cdot - 3) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.4

Simplify each term.

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Step 1.1.8.4.1

Multiply $x$ by $x$ .

$x^{2} + 18 = A (x^{2} + 3 \cdot - 3) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.4.2

Multiply $3$ by $- 3$ .

$x^{2} + 18 = A (x^{2} - 9) + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.5

Apply the distributive property.

$x^{2} + 18 = A x^{2} + A \cdot - 9 + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.6

Move $- 9$ to the left of $A$ .

$x^{2} + 18 = A x^{2} - 9 \cdot A + \frac{(B) (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.7

Cancel the common factor of $x + 3$ .

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Step 1.1.8.7.1

Cancel the common factor.

$x^{2} + 18 = A x^{2} - 9 A + \frac{B (x (x + 3) (x - 3))}{x + 3} + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.7.2

Divide $(B) (x (x - 3))$ by $1$ .

$x^{2} + 18 = A x^{2} - 9 A + (B) (x (x - 3)) + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.8

Apply the distributive property.

$x^{2} + 18 = A x^{2} - 9 A + B (x \cdot x + x \cdot - 3) + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.9

Multiply $x$ by $x$ .

$x^{2} + 18 = A x^{2} - 9 A + B (x^{2} + x \cdot - 3) + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.10

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

$x^{2} + 18 = A x^{2} - 9 A + B (x^{2} - 3 \cdot x) + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.11

Apply the distributive property.

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} + B (- 3 x) + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.12

Rewrite using the commutative property of multiplication.

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + \frac{(C) (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.13

Cancel the common factor of $x - 3$ .

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Step 1.1.8.13.1

Cancel the common factor.

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + \frac{C (x (x + 3) (x - 3))}{x - 3}$

Step 1.1.8.13.2

Divide $(C) (x (x + 3))$ by $1$ .

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + (C) (x (x + 3))$

Step 1.1.8.14

Apply the distributive property.

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + C (x \cdot x + x \cdot 3)$

Step 1.1.8.15

Multiply $x$ by $x$ .

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + C (x^{2} + x \cdot 3)$

Step 1.1.8.16

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

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + C (x^{2} + 3 \cdot x)$

Step 1.1.8.17

Apply the distributive property.

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + C x^{2} + C (3 x)$

Step 1.1.8.18

Rewrite using the commutative property of multiplication.

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 B x + C x^{2} + 3 C x$

Step 1.1.9

Simplify the expression.

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Step 1.1.9.1

Move $B$ .

$x^{2} + 18 = A x^{2} - 9 A + B x^{2} - 3 x B + C x^{2} + 3 C x$

Step 1.1.9.2

Move $- 9 A$ .

$x^{2} + 18 = A x^{2} + B x^{2} - 3 x B + C x^{2} + 3 C x - 9 A$

Step 1.1.9.3

Move $- 3 x B$ .

$x^{2} + 18 = A x^{2} + B x^{2} + C x^{2} - 3 x B + 3 C x - 9 A$

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 $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A + B + C$

Step 1.2.2

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.

$0 = - 3 B + 3 C$

Step 1.2.3

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.

$18 = - 9 A$

Step 1.2.4

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

$1 = A + B + C$

$0 = - 3 B + 3 C$

$18 = - 9 A$

$1 = A + B + C$

$0 = - 3 B + 3 C$

$18 = - 9 A$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Solve for $A$ in $18 = - 9 A$ .

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Step 1.3.1.1

Rewrite the equation as $- 9 A = 18$ .

$- 9 A = 18$

$1 = A + B + C$

$0 = - 3 B + 3 C$

Step 1.3.1.2

Divide each term in $- 9 A = 18$ by $- 9$ and simplify.

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Step 1.3.1.2.1

Divide each term in $- 9 A = 18$ by $- 9$ .

$\frac{- 9 A}{- 9} = \frac{18}{- 9}$

$1 = A + B + C$

$0 = - 3 B + 3 C$

Step 1.3.1.2.2

Simplify the left side.

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Step 1.3.1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.3.1.2.2.1.1

Cancel the common factor.

$\frac{- 9 A}{- 9} = \frac{18}{- 9}$

$1 = A + B + C$

$0 = - 3 B + 3 C$

Step 1.3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{18}{- 9}$

$1 = A + B + C$

$0 = - 3 B + 3 C$

$A = \frac{18}{- 9}$

$1 = A + B + C$

$0 = - 3 B + 3 C$

$A = \frac{18}{- 9}$

$1 = A + B + C$

$0 = - 3 B + 3 C$

Step 1.3.1.2.3

Simplify the right side.

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Step 1.3.1.2.3.1

Divide $18$ by $- 9$ .

$A = - 2$

$1 = A + B + C$

$0 = - 3 B + 3 C$

$A = - 2$

$1 = A + B + C$

$0 = - 3 B + 3 C$

$A = - 2$

$1 = A + B + C$

$0 = - 3 B + 3 C$

$A = - 2$

$1 = A + B + C$

$0 = - 3 B + 3 C$

Step 1.3.2

Replace all occurrences of $A$ with $- 2$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $A$ in $1 = A + B + C$ with $- 2$ .

$1 = (- 2) + B + C$

$A = - 2$

$0 = - 3 B + 3 C$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Remove parentheses.

$1 = - 2 + B + C$

$A = - 2$

$0 = - 3 B + 3 C$

$1 = - 2 + B + C$

$A = - 2$

$0 = - 3 B + 3 C$

$1 = - 2 + B + C$

$A = - 2$

$0 = - 3 B + 3 C$

Step 1.3.3

Solve for $B$ in $1 = - 2 + B + C$ .

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Step 1.3.3.1

Rewrite the equation as $- 2 + B + C = 1$ .

$- 2 + B + C = 1$

$A = - 2$

$0 = - 3 B + 3 C$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Add $2$ to both sides of the equation.

$B + C = 1 + 2$

$A = - 2$

$0 = - 3 B + 3 C$

Step 1.3.3.2.2

Subtract $C$ from both sides of the equation.

$B = 1 + 2 - C$

$A = - 2$

$0 = - 3 B + 3 C$

Step 1.3.3.2.3

Add $1$ and $2$ .

$B = 3 - C$

$A = - 2$

$0 = - 3 B + 3 C$

$B = 3 - C$

$A = - 2$

$0 = - 3 B + 3 C$

$B = 3 - C$

$A = - 2$

$0 = - 3 B + 3 C$

Step 1.3.4

Replace all occurrences of $B$ with $3 - C$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $B$ in $0 = - 3 B + 3 C$ with $3 - C$ .

$0 = - 3 (3 - C) + 3 C$

$B = 3 - C$

$A = - 2$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $- 3 (3 - C) + 3 C$ .

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Step 1.3.4.2.1.1

Simplify each term.

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Step 1.3.4.2.1.1.1

Apply the distributive property.

$0 = - 3 \cdot 3 - 3 (- C) + 3 C$

$B = 3 - C$

$A = - 2$

Step 1.3.4.2.1.1.2

Multiply $- 3$ by $3$ .

$0 = - 9 - 3 (- C) + 3 C$

$B = 3 - C$

$A = - 2$

Step 1.3.4.2.1.1.3

Multiply $- 1$ by $- 3$ .

$0 = - 9 + 3 C + 3 C$

$B = 3 - C$

$A = - 2$

$0 = - 9 + 3 C + 3 C$

$B = 3 - C$

$A = - 2$

Step 1.3.4.2.1.2

Add $3 C$ and $3 C$ .

$0 = - 9 + 6 C$

$B = 3 - C$

$A = - 2$

$0 = - 9 + 6 C$

$B = 3 - C$

$A = - 2$

$0 = - 9 + 6 C$

$B = 3 - C$

$A = - 2$

$0 = - 9 + 6 C$

$B = 3 - C$

$A = - 2$

Step 1.3.5

Solve for $C$ in $0 = - 9 + 6 C$ .

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Step 1.3.5.1

Rewrite the equation as $- 9 + 6 C = 0$ .

$- 9 + 6 C = 0$

$B = 3 - C$

$A = - 2$

Step 1.3.5.2

Add $9$ to both sides of the equation.

$6 C = 9$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3

Divide each term in $6 C = 9$ by $6$ and simplify.

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Step 1.3.5.3.1

Divide each term in $6 C = 9$ by $6$ .

$\frac{6 C}{6} = \frac{9}{6}$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3.2

Simplify the left side.

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Step 1.3.5.3.2.1

Cancel the common factor of $6$ .

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Step 1.3.5.3.2.1.1

Cancel the common factor.

$\frac{6 C}{6} = \frac{9}{6}$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{9}{6}$

$B = 3 - C$

$A = - 2$

$C = \frac{9}{6}$

$B = 3 - C$

$A = - 2$

$C = \frac{9}{6}$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3.3

Simplify the right side.

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Step 1.3.5.3.3.1

Cancel the common factor of $9$ and $6$ .

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Step 1.3.5.3.3.1.1

Factor $3$ out of $9$ .

$C = \frac{3 (3)}{6}$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3.3.1.2

Cancel the common factors.

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Step 1.3.5.3.3.1.2.1

Factor $3$ out of $6$ .

$C = \frac{3 \cdot 3}{3 \cdot 2}$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3.3.1.2.2

Cancel the common factor.

$C = \frac{3 \cdot 3}{3 \cdot 2}$

$B = 3 - C$

$A = - 2$

Step 1.3.5.3.3.1.2.3

Rewrite the expression.

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

$B = 3 - C$

$A = - 2$

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

$B = 3 - C$

$A = - 2$

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

$B = 3 - C$

$A = - 2$

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

$B = 3 - C$

$A = - 2$

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

$B = 3 - C$

$A = - 2$

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

$B = 3 - C$

$A = - 2$

Step 1.3.6

Replace all occurrences of $C$ with $\frac{3}{2}$ in each equation.

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Step 1.3.6.1

Replace all occurrences of $C$ in $B = 3 - C$ with $\frac{3}{2}$ .

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

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

$A = - 2$

Step 1.3.6.2

Simplify the right side.

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Step 1.3.6.2.1

Simplify $3 - (\frac{3}{2})$ .

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Step 1.3.6.2.1.1

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

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

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

$A = - 2$

Step 1.3.6.2.1.2

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

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

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

$A = - 2$

Step 1.3.6.2.1.3

Combine the numerators over the common denominator.

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

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

$A = - 2$

Step 1.3.6.2.1.4

Simplify the numerator.

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Step 1.3.6.2.1.4.1

Multiply $3$ by $2$ .

$B = \frac{6 - 3}{2}$

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

$A = - 2$

Step 1.3.6.2.1.4.2

Subtract $3$ from $6$ .

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

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

$A = - 2$

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

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

$A = - 2$

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

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

$A = - 2$

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

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

$A = - 2$

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

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

$A = - 2$

Step 1.3.7

List all of the solutions.

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

Step 1.4

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

$- \frac{2}{x} + \frac{\frac{3}{2}}{x + 3} + \frac{\frac{3}{2}}{x - 3}$

Step 1.5

Simplify.

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Step 1.5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.2

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

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

Step 1.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.4

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Let $u_{1} = x + 3$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 8.1

Let $u_{1} = x + 3$ . Find $\frac{d u_{1}}{d x}$ .

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Step 8.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 8.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 8.1.3

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

$1 + \frac{d}{d x} [3]$

Step 8.1.4

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

$1 + 0$

Step 8.1.5

Add $1$ and $0$ .

$1$

Step 8.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

Let $u_{2} = x - 3$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 11.1

Let $u_{2} = x - 3$ . Find $\frac{d u_{2}}{d x}$ .

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Step 11.1.1

Differentiate $x - 3$ .

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

Step 11.1.2

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

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

Step 11.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 11.1.4

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

$1 + 0$

Step 11.1.5

Add $1$ and $0$ .

$1$

Step 11.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 12

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

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

Step 13

Simplify.

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

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{1}$ with $x + 3$ .

$- 2 \ln (| x |) + \frac{3}{2} \ln (| x + 3 |) + \frac{3}{2} \ln (| u_{2} |) + C$

Step 14.2

Replace all occurrences of $u_{2}$ with $x - 3$ .

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