Here,
#I=int(2x+3)/(x^4-9x^2)dx=int(2x+3)/(x^2(x^2-9))dx#
#I=int(2x+3)/(x^2(x-3)(x+3))dx#
Let,
#(2x+3)/(x^2(x-3)(x+3))=A/x+B/x^2+C/(x-3)+D/(x+3)#
#2x+3#=#Ax(x^2-9)+B(x^2-9)+Cx^2(x+3)+Dx^2(x-3)#
#2x+3#=#x^3(A+C+D)+x^2(B+3C-3D)+x(-9A)-9B#
#"Comparing co efficients of "# #x^3,x^2,x and# #"constant term
:"#
#A+C+D=0..........to(1)#
#B+3C-3D=0.......to(2)#
#-9A=2=>color(red)(A=-2/9to(3)#
#-9B=3=>color(red)(B=-1/3to(4)#
From #(1) and (3)color(white)(..........)#From #(2) and(4)#
#C+D=2/9to(5)color(white)(.......)3C-3D=1/3=>C-D=1/9to(6)#
Adding #(5) and (6)# we get
#C+C=2/9+1/9=>2C=3/9=>C=3/18=>color(red)(C=1/6#
From #(5)# ,we get
#1/6+D=2/9=>D=2/9-1/6=(4-3)/18=>color(red)(D=1/18#
So,
#I=int[(-2/9)/x+(-1/3)/x^2+(1/6)/(x-3)+(1/18)/(x+3)]dx#
#=-2/9ln|x|-1/3(x^(-2+1)/(-2+1))+1/6ln|x-3|+1/18ln|x+3|# + c
#=-2/9ln|x|-1/3(x^(-1)/(-1))+1/6ln|x-3|+1/18ln|x+3|+c#
#=1/18ln|x+3|+1/6ln|x-3|-2/9ln|x|+1/(3x)+c#
