The partial fraction equation is:
#1 /(x(x+1)^2) = A/x + B/(x+1)+C/(x+1)^2#
Multiply both sides by #x(x+1)^2#
#1 = A(x+1)^2 + Bx(x+1)+Cx#
Eliminate B and C by Letting #x = 0#:
#1 = A(0+1)^2#
#A = 1#
#1 = (x+1)^2 + Bx(x+1)+Cx#
Eliminate B by letting #x = -1#
#1 = C(-1)#
#C = -1#
#1 = (x+1)^2 + Bx(x+1)-x#
Let #x = 1#:
#1 = (1+1)^2 + B(1)(1+1)-1#
#2-4 = 2B#
#B = -1#
The partial fraction expansion is:
#1 /(x(x+1)^2) = 1/x -1/(x+1)-1/(x+1)^2#
Check:
#1/x -1/(x+1)-1/(x+1)^2 = 1/x(x+1)^2/(x+1)^2-1/(x+1)(x(x+1))/(x(x+1)) - 1/(x+1)^2
x/x#
#1/x -1/(x+1)-1/(x+1)^2 = (x^2+ 2x +1-x^2-x -x)/(x(x+1)^2)#
#1/x -1/(x+1)-1/(x+1)^2 = 1/(x(x+1)^2)#
This checks.
The original integrand is equal to the partial fractions:
#int ( (dx) / ( x(x+1)^2 ) ) = int 1/x -1/(x+1)-1/(x+1)^2 dx#
Separate into 3 integrals:
#int ( (dx) / ( x(x+1)^2 ) ) = int 1/x dx - int 1/(x+1) dx - int 1/(x+1)^2 dx#
These integrals are well known:
#int ( (dx) / ( x(x+1)^2 ) ) = ln|x| - ln|x+1| + 1/(x+1) + C#
