How do you use partial fraction decomposition to decompose the fraction to integrate #26/(6x^2+5x-6)#?

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Answer 1 · 2018-06-08T16:37:18

See process below

First of all, we factorize the polynomial #6x^2+5x-6#

#6x^2+5x-6=0#

#x=(-5+-sqrt(25+144))/12=(-5+-13)/6# this give two solutions

#x_1=-3# and #x_2=4/3#

Thus we have #6x^2+5x-6=(x+3)(x-4/3)#

#1/(6x^2+5x-6)=A/(x+3)+B/(x-4/3)# transposing terms and equalizing we have

#A+B=0#
#3B-4/3A=1# from here we find #A=-3/13# and #B=3/13#

Then finally we have #1/(6x^2+5x-6)=(-3/13)/(x+3)+(3/13)/(x-4/3)#