How do you integrate (5x)/(2x^2+11x+12)5x2x2+11x+12 using partial fractions?
1 Answer
Explanation:
We start by factoring the denominator.
The partial fraction decomposition will therefore be of the form:
A/(2x + 3) + B/(x + 4) = (5x)/((2x + 3)(x + 4)A2x+3+Bx+4=5x(2x+3)(x+4)
A(x + 4) + B(2x + 3) = 5xA(x+4)+B(2x+3)=5x
Ax + 4A + 2Bx+ 3B = 5xAx+4A+2Bx+3B=5x
(A + 2B)x + (4A + 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
int(4/(x + 4) - 3/(2x + 3))dx
We know that
int(4/(x + 4) - 3/(2x + 3))dx = 4ln|x + 4| - 3/2ln|2x + 3| + C
Hopefully this helps!