How do you integrate #2 ln (x-5)#?

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Answer 1 · 2016-06-21T10:08:09

#color(blue)(int 2 ln(x-5) dx=(2x-10)*ln(x-5)-2x +C)#

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.