Set up the equation to solve for the variables A, B,C
#int (4x^2+6x-2)/((x-1)(x+1)^2) dx=int (A/(x-1 )+B/(x+1)+C/(x+1)^2)dx#
Let us solve for A, B, C first
#(4x^2+6x-2)/((x-1)(x+1)^2) =A/(x-1 )+B/(x+1)+C/(x+1)^2#
LCD #=(x-1)(x+1)^2#
#(4x^2+6x-2)/((x-1)(x+1)^2) =(A(x+1)^2+B(x^2-1)+C(x-1))/((x-1)(x+1)^2)#
Simplify
#(4x^2+6x-2)/((x-1)(x+1)^2) =(A(x^2+2x+1)+B(x^2-1)+C(x-1))/((x-1)(x+1)^2)#
#(4x^2+6x-2)/((x-1)(x+1)^2) =(Ax^2+2Ax+A+Bx^2-B+Cx-C)/((x-1)(x+1)^2)#
Rearrange the terms of the right side
#(4x^2+6x-2)/((x-1)(x+1)^2) =(Ax^2+Bx^2+2Ax+Cx+A-B-C)/((x-1)(x+1)^2)#
let us set up the equations to solve for A, B, C by matching the numerical coefficients of left and right terms
#A+B=4" "#first equation
#2A+C=6" "#second equation
#A-B-C=-2" "#third equation
Simultaneous solution using second and third equation results to
#2A+A+C-C-B=6-2#
#3A-B=4" "#fourth equation
Using now the first and the fourth equations
#3A-B=4" "#fourth equation
#3(4-B)-B=4" "#fourth equation
#12-3B-B=4#
#-4B=4-12#
#-4B=-8#
#B=2#
Solve for A using #3A-B=4" "#fourth equation
#3A-2=4" "#fourth equation
#3A=4+2#
#3A=6#
#A=2#
Solve C using the #2A+C=6" "#second equation and #A=2# and #B=2#
#2A+C=6" "#second equation
#2(2)+C=6#
#4+C=6#
#C=6-4#
#C=2#
We now perform our integration
#int (4x^2+6x-2)/((x-1)(x+1)^2) dx=int (2/(x-1 )+2/(x+1)+2/(x+1)^2)dx#
#int (4x^2+6x-2)/((x-1)(x+1)^2) dx=int (2/(x-1 )+2/(x+1)+2*(x+1)^(-2))dx#
#int (4x^2+6x-2)/((x-1)(x+1)^2) dx=2ln(x-1 )+2ln(x+1)+(2*(x+1)^(-2+1))/(-2+1)+C_o#
#int (4x^2+6x-2)/((x-1)(x+1)^2) dx=2ln(x-1 )+2ln(x+1)-2/(x+1)+C_o#
God bless.....I hope the explanation is useful.
