How do you use partial fraction decomposition to decompose the fraction to integrate #(x^2+x)/((x+2)(x-1)^2)#?

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Answer 1 · 2015-08-23T21:00:46

#(x^2+x)/((x+2)(x-1)^2) = (2/9)/(x+2) + (7/9)/(x-1) + (2/3)/(x-1)^2 #:

#(x^2+x)/((x+2)(x-1)^2)#

We want

#A/(x+2) + B/(x-1) + C/(x-1)^2 = (x^2+x)/((x+2)(x-1)^2) #:

Clear the denominator to get:

#A(x^2-2x+1)+B(x^2+x-2)+C(x+2) = x^2+x#

So we need to solve the system:

#A+B=1#
#-2A+B+C=1#
#A-2B+2C=0#

Solve to get:

#A = 2/9#, #" "B = 7/9#, and #" "C=2/3#