Calculus Examples

$\int \frac{6 x^{2} - 8 x + 6}{3 x^{3} - 6 x^{2} + 9 x} d x$

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 $2$ out of $6 x^{2} - 8 x + 6$ .

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Step 1.1.1.1.1

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

$\frac{2 (3 x^{2}) - 8 x + 6}{3 x^{3} - 6 x^{2} + 9 x}$

Step 1.1.1.1.2

Factor $2$ out of $- 8 x$ .

$\frac{2 (3 x^{2}) + 2 (- 4 x) + 6}{3 x^{3} - 6 x^{2} + 9 x}$

Step 1.1.1.1.3

Factor $2$ out of $6$ .

$\frac{2 (3 x^{2}) + 2 (- 4 x) + 2 (3)}{3 x^{3} - 6 x^{2} + 9 x}$

Step 1.1.1.1.4

Factor $2$ out of $2 (3 x^{2}) + 2 (- 4 x)$ .

$\frac{2 (3 x^{2} - 4 x) + 2 (3)}{3 x^{3} - 6 x^{2} + 9 x}$

Step 1.1.1.1.5

Factor $2$ out of $2 (3 x^{2} - 4 x) + 2 (3)$ .

$\frac{2 (3 x^{2} - 4 x + 3)}{3 x^{3} - 6 x^{2} + 9 x}$

Step 1.1.1.2

Factor $3 x$ out of $3 x^{3} - 6 x^{2} + 9 x$ .

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Step 1.1.1.2.1

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

$\frac{2 (3 x^{2} - 4 x + 3)}{3 x (x^{2}) - 6 x^{2} + 9 x}$

Step 1.1.1.2.2

Factor $3 x$ out of $- 6 x^{2}$ .

$\frac{2 (3 x^{2} - 4 x + 3)}{3 x (x^{2}) + 3 x (- 2 x) + 9 x}$

Step 1.1.1.2.3

Factor $3 x$ out of $9 x$ .

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

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

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

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 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.

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

Step 1.1.3

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

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

Step 1.1.4

Cancel the common factor of $3$ .

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Step 1.1.4.1

Cancel the common factor.

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

Step 1.1.4.2

Rewrite the expression.

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

Step 1.1.5

Cancel the common factor of $x$ .

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Step 1.1.5.1

Cancel the common factor.

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

Step 1.1.5.2

Rewrite the expression.

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

Step 1.1.6

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

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Step 1.1.6.1

Cancel the common factor.

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

Step 1.1.6.2

Divide $2 (3 x^{2} - 4 x + 3)$ by $1$ .

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

Step 1.1.7

Apply the distributive property.

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

Step 1.1.8

Simplify.

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Step 1.1.8.1

Multiply $3$ by $2$ .

$6 x^{2} + 2 (- 4 x) + 2 \cdot 3 = \frac{A (3 x (x^{2} - 2 x + 3))}{x} + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.8.2

Multiply $- 4$ by $2$ .

$6 x^{2} - 8 x + 2 \cdot 3 = \frac{A (3 x (x^{2} - 2 x + 3))}{x} + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.8.3

Multiply $2$ by $3$ .

$6 x^{2} - 8 x + 6 = \frac{A (3 x (x^{2} - 2 x + 3))}{x} + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9

Simplify each term.

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Step 1.1.9.1

Cancel the common factor of $x$ .

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Step 1.1.9.1.1

Cancel the common factor.

$6 x^{2} - 8 x + 6 = \frac{A (3 x (x^{2} - 2 x + 3))}{x} + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.1.2

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

$6 x^{2} - 8 x + 6 = A (3 (x^{2} - 2 x + 3)) + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.2

Rewrite using the commutative property of multiplication.

$6 x^{2} - 8 x + 6 = 3 A (x^{2} - 2 x + 3) + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.3

Apply the distributive property.

$6 x^{2} - 8 x + 6 = 3 A x^{2} + 3 A (- 2 x) + 3 A \cdot 3 + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.4

Simplify.

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Step 1.1.9.4.1

Rewrite using the commutative property of multiplication.

$6 x^{2} - 8 x + 6 = 3 A x^{2} + 3 \cdot (- 2 A x) + 3 A \cdot 3 + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.4.2

Multiply $3$ by $3$ .

$6 x^{2} - 8 x + 6 = 3 A x^{2} + 3 \cdot (- 2 A x) + 9 A + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.5

Multiply $3$ by $- 2$ .

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.6

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

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Step 1.1.9.6.1

Cancel the common factor.

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + \frac{(B x + C) (3 x (x^{2} - 2 x + 3))}{x^{2} - 2 x + 3}$

Step 1.1.9.6.2

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

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + (B x + C) (3 x)$

Step 1.1.9.7

Apply the distributive property.

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + B x (3 x) + C (3 x)$

Step 1.1.9.8

Rewrite using the commutative property of multiplication.

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + 3 B x \cdot x + C (3 x)$

Step 1.1.9.9

Rewrite using the commutative property of multiplication.

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + 3 B x \cdot x + 3 C x$

Step 1.1.9.10

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

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Step 1.1.9.10.1

Move $x$ .

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + 3 B (x \cdot x) + 3 C x$

Step 1.1.9.10.2

Multiply $x$ by $x$ .

$6 x^{2} - 8 x + 6 = 3 A x^{2} - 6 A x + 9 A + 3 B x^{2} + 3 C x$

Step 1.1.10

Reorder.

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Step 1.1.10.1

Move $A$ .

$6 x^{2} - 8 x + 6 = 3 x^{2} A - 6 A x + 9 A + 3 B x^{2} + 3 C x$

Step 1.1.10.2

Move $A$ .

$6 x^{2} - 8 x + 6 = 3 x^{2} A - 6 x A + 9 A + 3 B x^{2} + 3 C x$

Step 1.1.10.3

Move $B$ .

$6 x^{2} - 8 x + 6 = 3 x^{2} A - 6 x A + 9 A + 3 x^{2} B + 3 C x$

Step 1.1.10.4

Move $9 A$ .

$6 x^{2} - 8 x + 6 = 3 x^{2} A - 6 x A + 3 x^{2} B + 3 C x + 9 A$

Step 1.1.10.5

Move $- 6 x A$ .

$6 x^{2} - 8 x + 6 = 3 x^{2} A + 3 x^{2} B - 6 x A + 3 C x + 9 A$

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.

$6 = 3 A + 3 B$

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.

$- 8 = - 6 A + 3 C$

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.

$6 = 9 A$

Step 1.2.4

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

$6 = 9 A$

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

$6 = 9 A$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Solve for $A$ in $6 = 9 A$ .

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Step 1.3.1.1

Rewrite the equation as $9 A = 6$ .

$9 A = 6$

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.1.2

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

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Step 1.3.1.2.1

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

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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.

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.1.2.2.1.2

Divide $A$ by $1$ .

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.1.2.3

Simplify the right side.

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Step 1.3.1.2.3.1

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

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Step 1.3.1.2.3.1.1

Factor $3$ out of $6$ .

$A = \frac{3 (2)}{9}$

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.1.2.3.1.2

Cancel the common factors.

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Step 1.3.1.2.3.1.2.1

Factor $3$ out of $9$ .

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.1.2.3.1.2.2

Cancel the common factor.

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.1.2.3.1.2.3

Rewrite the expression.

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

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

$6 = 3 A + 3 B$

$- 8 = - 6 A + 3 C$

Step 1.3.2

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

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Step 1.3.2.1

Replace all occurrences of $A$ in $6 = 3 A + 3 B$ with $\frac{2}{3}$ .

$6 = 3 (\frac{2}{3}) + 3 B$

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

$- 8 = - 6 A + 3 C$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Cancel the common factor of $3$ .

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Step 1.3.2.2.1.1

Cancel the common factor.

$6 = 3 (\frac{2}{3}) + 3 B$

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

$- 8 = - 6 A + 3 C$

Step 1.3.2.2.1.2

Rewrite the expression.

$6 = 2 + 3 B$

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

$- 8 = - 6 A + 3 C$

$6 = 2 + 3 B$

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

$- 8 = - 6 A + 3 C$

$6 = 2 + 3 B$

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

$- 8 = - 6 A + 3 C$

Step 1.3.2.3

Replace all occurrences of $A$ in $- 8 = - 6 A + 3 C$ with $\frac{2}{3}$ .

$- 8 = - 6 (\frac{2}{3}) + 3 C$

$6 = 2 + 3 B$

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

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify each term.

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Step 1.3.2.4.1.1

Cancel the common factor of $3$ .

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Step 1.3.2.4.1.1.1

Factor $3$ out of $- 6$ .

$- 8 = 3 (- 2) (\frac{2}{3}) + 3 C$

$6 = 2 + 3 B$

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

Step 1.3.2.4.1.1.2

Cancel the common factor.

$- 8 = 3 \cdot (- 2 (\frac{2}{3})) + 3 C$

$6 = 2 + 3 B$

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

Step 1.3.2.4.1.1.3

Rewrite the expression.

$- 8 = - 2 \cdot 2 + 3 C$

$6 = 2 + 3 B$

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

$- 8 = - 2 \cdot 2 + 3 C$

$6 = 2 + 3 B$

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

Step 1.3.2.4.1.2

Multiply $- 2$ by $2$ .

$- 8 = - 4 + 3 C$

$6 = 2 + 3 B$

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

$- 8 = - 4 + 3 C$

$6 = 2 + 3 B$

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

$- 8 = - 4 + 3 C$

$6 = 2 + 3 B$

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

$- 8 = - 4 + 3 C$

$6 = 2 + 3 B$

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

Step 1.3.3

Solve for $C$ in $- 8 = - 4 + 3 C$ .

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Step 1.3.3.1

Rewrite the equation as $- 4 + 3 C = - 8$ .

$- 4 + 3 C = - 8$

$6 = 2 + 3 B$

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

Step 1.3.3.2

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

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Step 1.3.3.2.1

Add $4$ to both sides of the equation.

$3 C = - 8 + 4$

$6 = 2 + 3 B$

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

Step 1.3.3.2.2

Add $- 8$ and $4$ .

$3 C = - 4$

$6 = 2 + 3 B$

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

$3 C = - 4$

$6 = 2 + 3 B$

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

Step 1.3.3.3

Divide each term in $3 C = - 4$ by $3$ and simplify.

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Step 1.3.3.3.1

Divide each term in $3 C = - 4$ by $3$ .

$\frac{3 C}{3} = \frac{- 4}{3}$

$6 = 2 + 3 B$

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

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{3 C}{3} = \frac{- 4}{3}$

$6 = 2 + 3 B$

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

Step 1.3.3.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 4}{3}$

$6 = 2 + 3 B$

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

$C = \frac{- 4}{3}$

$6 = 2 + 3 B$

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

$C = \frac{- 4}{3}$

$6 = 2 + 3 B$

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

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Move the negative in front of the fraction.

$C = - \frac{4}{3}$

$6 = 2 + 3 B$

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

$C = - \frac{4}{3}$

$6 = 2 + 3 B$

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

$C = - \frac{4}{3}$

$6 = 2 + 3 B$

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

$C = - \frac{4}{3}$

$6 = 2 + 3 B$

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

Step 1.3.4

Replace all occurrences of $C$ with $- \frac{4}{3}$ in each equation.

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Step 1.3.4.1

Rewrite the equation as $2 + 3 B = 6$ .

$2 + 3 B = 6$

$C = - \frac{4}{3}$

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

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 $2$ from both sides of the equation.

$3 B = 6 - 2$

$C = - \frac{4}{3}$

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

Step 1.3.4.2.2

Subtract $2$ from $6$ .

$3 B = 4$

$C = - \frac{4}{3}$

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

$3 B = 4$

$C = - \frac{4}{3}$

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

Step 1.3.4.3

Divide each term in $3 B = 4$ by $3$ and simplify.

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Step 1.3.4.3.1

Divide each term in $3 B = 4$ by $3$ .

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

$C = - \frac{4}{3}$

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

Step 1.3.4.3.2

Simplify the left side.

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Step 1.3.4.3.2.1

Cancel the common factor of $3$ .

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Step 1.3.4.3.2.1.1

Cancel the common factor.

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

$C = - \frac{4}{3}$

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

Step 1.3.4.3.2.1.2

Divide $B$ by $1$ .

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

$C = - \frac{4}{3}$

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

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

$C = - \frac{4}{3}$

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

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

$C = - \frac{4}{3}$

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

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

$C = - \frac{4}{3}$

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

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

$C = - \frac{4}{3}$

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

Step 1.3.5

List all of the solutions.

$B = \frac{4}{3}, C = - \frac{4}{3}, A = \frac{2}{3}$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.5.1.1

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

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

Step 1.5.1.2

Combine.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Cancel the common factor of $3$ .

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Step 1.5.3.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

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

Step 1.5.3.2

Cancel the common factor.

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

Step 1.5.3.3

Rewrite the expression.

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

Step 1.5.4

Simplify the numerator.

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Step 1.5.4.1

Cancel the common factor of $3$ .

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Step 1.5.4.1.1

Cancel the common factor.

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

Step 1.5.4.1.2

Rewrite the expression.

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

Step 1.5.4.2

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

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Step 1.5.4.2.1

Factor $4$ out of $4 x$ .

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

Step 1.5.4.2.2

Factor $4$ out of $- 4$ .

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

Step 1.5.4.2.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

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

Step 1.5.5

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

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Step 1.5.5.1

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

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

Step 1.5.5.2

Factor $3$ out of $3 \cdot 3$ .

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

Step 1.5.5.3

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

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

Step 1.5.5.4

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

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

Step 1.5.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.7

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $x^{2} - 2 x + 3$ .

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

Step 6.1.2

Differentiate.

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 6.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 6.1.3.2

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

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

Step 6.1.3.3

Multiply $- 2$ by $1$ .

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

Step 6.1.4

Differentiate using the Constant Rule.

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Step 6.1.4.1

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

$2 x - 2 + 0$

Step 6.1.4.2

Add $2 x - 2$ and $0$ .

$2 x - 2$

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Multiply $2$ by $3$ .

$\frac{2}{3} (\ln (| x |) + C) + \frac{4}{6} \int \frac{1}{u} d u$

Step 9.3

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

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Step 9.3.1

Factor $2$ out of $4$ .

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

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $6$ .

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

Step 9.3.2.2

Cancel the common factor.

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

Step 9.3.2.3

Rewrite the expression.

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

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

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