The given
#int 2 ln(x-5) dx#
Let #u=ln (x-5)#
Let #dv=dx#
Let #v=x#
Let #du=(1/(x-5))*dx#
Using integration by parts
#int u*dv=uv-int v*du#
#int ln(x-5) dx=x*ln(x-5)-int x/(x-5)dx#
#int ln(x-5) dx=x*ln(x-5)-int (x-5+5)/(x-5)dx#
#int ln(x-5) dx=x*ln(x-5)-int (1+5/(x-5))dx#
#int ln(x-5) dx=x*ln(x-5)-x -5*ln(x-5)#
So that
#int 2 ln(x-5) dx=2*[x*ln(x-5)-x -5*ln(x-5)]#
#int 2 ln(x-5) dx=2x*ln(x-5)-2x -10*ln(x-5)+C#
#color(blue)(int 2 ln(x-5) dx=(2x-10)*ln(x-5)-2x +C)#
God bless....I hope the explanation is useful.