Trigonometry Examples

$\frac{13 x + 2}{{(3 x + 1)}^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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

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

Step 1.2

$\frac{A}{{(3 x + 1)}^{2}} + \frac{B}{3 x + 1}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(3 x + 1)}^{2}$ .

$\frac{(13 x + 2) {(3 x + 1)}^{2}}{{(3 x + 1)}^{2}} = \frac{(A) {(3 x + 1)}^{2}}{{(3 x + 1)}^{2}} + \frac{(B) {(3 x + 1)}^{2}}{3 x + 1}$

Step 1.4

Cancel the common factor of ${(3 x + 1)}^{2}$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(13 x + 2) {(3 x + 1)}^{2}}{{(3 x + 1)}^{2}} = \frac{(A) {(3 x + 1)}^{2}}{{(3 x + 1)}^{2}} + \frac{(B) {(3 x + 1)}^{2}}{3 x + 1}$

Step 1.4.2

Divide $13 x + 2$ by $1$ .

$13 x + 2 = \frac{(A) {(3 x + 1)}^{2}}{{(3 x + 1)}^{2}} + \frac{(B) {(3 x + 1)}^{2}}{3 x + 1}$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of ${(3 x + 1)}^{2}$ .

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Step 1.5.1.1

Cancel the common factor.

$13 x + 2 = \frac{A {(3 x + 1)}^{2}}{{(3 x + 1)}^{2}} + \frac{(B) {(3 x + 1)}^{2}}{3 x + 1}$

Step 1.5.1.2

Divide $A$ by $1$ .

$13 x + 2 = A + \frac{(B) {(3 x + 1)}^{2}}{3 x + 1}$

Step 1.5.2

Cancel the common factor of ${(3 x + 1)}^{2}$ and $3 x + 1$ .

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Step 1.5.2.1

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

$13 x + 2 = A + \frac{(3 x + 1) (B (3 x + 1))}{3 x + 1}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Multiply by $1$ .

$13 x + 2 = A + \frac{(3 x + 1) (B (3 x + 1))}{(3 x + 1) \cdot 1}$

Step 1.5.2.2.2

Cancel the common factor.

$13 x + 2 = A + \frac{(3 x + 1) (B (3 x + 1))}{(3 x + 1) \cdot 1}$

Step 1.5.2.2.3

Rewrite the expression.

$13 x + 2 = A + \frac{B (3 x + 1)}{1}$

Step 1.5.2.2.4

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

$13 x + 2 = A + B (3 x + 1)$

Step 1.5.3

Apply the distributive property.

$13 x + 2 = A + B (3 x) + B \cdot 1$

Step 1.5.4

Rewrite using the commutative property of multiplication.

$13 x + 2 = A + 3 B x + B \cdot 1$

Step 1.5.5

Multiply $B$ by $1$ .

$13 x + 2 = A + 3 B x + B$

Step 1.6

Simplify the expression.

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Step 1.6.1

Move $B$ .

$13 x + 2 = A + 3 x B + B$

Step 1.6.2

Reorder $A$ and $3 x B$ .

$13 x + 2 = 3 x B + A + B$

Step 2

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

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Step 2.1

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.

$13 = 3 B$

Step 2.2

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.

$2 = A + B$

Step 2.3

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

$13 = 3 B$

$2 = A + B$

$13 = 3 B$

$2 = A + B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $B$ in $13 = 3 B$ .

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Step 3.1.1

Rewrite the equation as $3 B = 13$ .

$3 B = 13$

$2 = A + B$

Step 3.1.2

Divide each term in $3 B = 13$ by $3$ and simplify.

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Step 3.1.2.1

Divide each term in $3 B = 13$ by $3$ .

$\frac{3 B}{3} = \frac{13}{3}$

$2 = A + B$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{3 B}{3} = \frac{13}{3}$

$2 = A + B$

Step 3.1.2.2.1.2

Divide $B$ by $1$ .

$B = \frac{13}{3}$

$2 = A + B$

$B = \frac{13}{3}$

$2 = A + B$

$B = \frac{13}{3}$

$2 = A + B$

$B = \frac{13}{3}$

$2 = A + B$

$B = \frac{13}{3}$

$2 = A + B$

Step 3.2

Replace all occurrences of $B$ with $\frac{13}{3}$ in each equation.

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Step 3.2.1

Replace all occurrences of $B$ in $2 = A + B$ with $\frac{13}{3}$ .

$2 = A + \frac{13}{3}$

$B = \frac{13}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Remove parentheses.

$2 = A + \frac{13}{3}$

$B = \frac{13}{3}$

$2 = A + \frac{13}{3}$

$B = \frac{13}{3}$

$2 = A + \frac{13}{3}$

$B = \frac{13}{3}$

Step 3.3

Solve for $A$ in $2 = A + \frac{13}{3}$ .

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Step 3.3.1

Rewrite the equation as $A + \frac{13}{3} = 2$ .

$A + \frac{13}{3} = 2$

$B = \frac{13}{3}$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $\frac{13}{3}$ from both sides of the equation.

$A = 2 - \frac{13}{3}$

$B = \frac{13}{3}$

Step 3.3.2.2

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

$A = 2 \cdot \frac{3}{3} - \frac{13}{3}$

$B = \frac{13}{3}$

Step 3.3.2.3

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

$A = \frac{2 \cdot 3}{3} - \frac{13}{3}$

$B = \frac{13}{3}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$A = \frac{2 \cdot 3 - 13}{3}$

$B = \frac{13}{3}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $2$ by $3$ .

$A = \frac{6 - 13}{3}$

$B = \frac{13}{3}$

Step 3.3.2.5.2

Subtract $13$ from $6$ .

$A = \frac{- 7}{3}$

$B = \frac{13}{3}$

$A = \frac{- 7}{3}$

$B = \frac{13}{3}$

Step 3.3.2.6

Move the negative in front of the fraction.

$A = - \frac{7}{3}$

$B = \frac{13}{3}$

$A = - \frac{7}{3}$

$B = \frac{13}{3}$

$A = - \frac{7}{3}$

$B = \frac{13}{3}$

Step 3.4

Solve the system of equations.

$A = - \frac{7}{3} B = \frac{13}{3}$

Step 3.5

List all of the solutions.

$A = - \frac{7}{3}, B = \frac{13}{3}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{{(3 x + 1)}^{2}} + \frac{B}{3 x + 1}$ with the values found for $A$ and $B$ .

$\frac{- \frac{7}{3}}{{(3 x + 1)}^{2}} + \frac{\frac{13}{3}}{3 x + 1}$