How do you integrate #(5x)/(2x^2+11x+12)# using partial fractions?

Question

Aurora Alvarado · Accepted Answer

#int((5x)/(2x^2 + 11x + 12))dx = 4ln|x + 4| - 3/2ln|2x + 3| + C#

We start by factoring the denominator.

#2x^2 + 11x + 12 = 2x^2 + 8x + 3x + 12 = 2x(x + 4) + 3(x +4) = (2x + 3)(x + 4)#.

The partial fraction decomposition will therefore be of the form:

#A/(2x + 3) + B/(x + 4) = (5x)/((2x + 3)(x + 4)#

#A(x + 4) + B(2x + 3) = 5x#

#Ax + 4A + 2Bx+ 3B = 5x#

#(A + 2B)x + (4A + 3B) = 5x#

We now write a systems of equations.

#{(A + 2B= 5), (4A + 3B = 0):}#

Solve:

#A = 5 - 2B#

#4(5 - 2B) + 3B = 0#

#20 - 8B + 3B = 0#

#-5B = -20#

#B = 4#

#A + 2(4) = 5#

#A = -3#

Hence, the partial fraction decomposition is #4/(x + 4) - 3/(2x + 3)#. The integral becomes

#int(4/(x + 4) - 3/(2x + 3))dx#

We know that #int(1/u)du = ln|u| + C#. Therefore:

#int(4/(x + 4) - 3/(2x + 3))dx = 4ln|x + 4| - 3/2ln|2x + 3| + C#

Hopefully this helps!

Emily Ammerman · Answer

undefined To integrate the rational function (5x)/(2x^2 + 11x + 12) using partial fractions, follow these steps:

1. Factor the denominator: 2x^2 + 11x + 12 = (2x + 3)(x + 4).
2. Decompose the rational function into partial fractions: (5x)/(2x^2 + 11x + 12) = A/(2x + 3) + B/(x + 4).
3. Multiply both sides by the denominator to clear the fractions: 5x = A(x + 4) + B(2x + 3).
4. Expand and collect like terms: 5x = (A + 2B)x + (4A + 3B).
5. Equate the coefficients of like terms on both sides to find the values of A and B.
6. Solve the system of equations to find the values of A and B.
7. Once you have A and B, integrate each term separately.
8. The integral of (5x)/(2x^2 + 11x + 12) is A ln|2x + 3| + B ln|x + 4| + C, where C is the constant of integration.

After solving for A and B, the integral would be:
A ln|2x + 3| + B ln|x + 4| + C.