How do you use partial fraction decomposition to decompose the fraction to integrate #(13x) / (6x^2 + 5x - 6)#?

Question

Luke Alvarez · Accepted Answer

Using a fraction whose denominator is quadratic, you need to check whether or not it is factorable. #6x^2 + 5x - 6 = (2x+3)(3x-2)# Since it is, we only need to write out the following: #int (13x)/((2x + 3)(3x - 2))dx = int A/(2x + 3) + B/(3x - 2)dx# To find what #A# and #B# are, you could start by cross-multiplying. For now, ignore the integral sign and #dx#. #= (A(3x-2) + B(2x+3))/((2x+3)(3x - 2))# #= (3Ax - 2A + 2Bx+3B)/((2x+3)(3x - 2))# #= ((3A + 2B)x + (-2A + 3B))/((2x+3)(3x - 2))# Thus, we have the equations: #-2A + 3B = 0 => B = 2/3 A# #3A + 2B = 13 => 3A + 4/3 A = 13#
#13/3 A = 13#
#:. color(green)(A = 3) => color(green)(B = 2)# At this point we are pretty much done. #= int 3/(2x + 3) + 2/(3x - 2)dx# #= color(blue)(3/2ln|2x + 3| + 2/3ln|3x - 2| + C)#

Aurora Alvarado · Answer

undefined To decompose the fraction $ \frac{13x}{6x^2 + 5x - 6} $ using partial fraction decomposition, follow these steps:

1. Factor the denominator $ 6x^2 + 5x - 6 $ into irreducible factors.
2. Write the original fraction as the sum of simpler fractions with unknown constants as numerators.
3. Equate the original fraction to the sum of the simpler fractions and solve for the unknown constants.
4. Integrate each simpler fraction.

Here are the detailed steps:

1. Factor the denominator $ 6x^2 + 5x - 6 $:
   $$ 6x^2 + 5x - 6 = (2x - 3)(3x + 2) $$

2. Write the original fraction as the sum of simpler fractions:
   $$ \frac{13x}{6x^2 + 5x - 6} = \frac{A}{2x - 3} + \frac{B}{3x + 2} $$

3. Equate the original fraction to the sum of the simpler fractions:
   $$ \frac{13x}{6x^2 + 5x - 6} = \frac{A}{2x - 3} + \frac{B}{3x + 2} $$
   Multiply both sides by the denominator $ (2x - 3)(3x + 2) $:
   $$ 13x = A(3x + 2) + B(2x - 3) $$

4. Solve for $ A $ and $ B $ by equating coefficients:
   $$ 13x = (3A + 2B)x + (2A - 3B) $$
   Compare coefficients of like terms:
   $$ 3A + 2B = 13 $$ (coefficients of $ x $)
   $$ 2A - 3B = 0 $$ (constant terms)

Solve this system of equations to find $ A $ and $ B $.

5. Once you have found the values of $ A $ and $ B $, integrate each simpler fraction separately.

That's the process to decompose the fraction and integrate it using partial fraction decomposition.