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.

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