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

1 Answer
Jun 5, 2015

2int 1*ln(x-5)dx

You can integrate this using integration by parts. The general formula for integration by parts is:

uv - intvdu

Now for w-substitution, let
x-5 = w
x = w+5
dw = dx

2intln(x-5)dx = 2intlnwdw

For integration by parts, let
u = lnw
dv = 1dw
v = w
du = 1/wdw

= 2[wln|w| - intw/wdw]

= 2[wln|w| - w + C]

= 2[(x-5)ln|x-5| - (x-5) + C]

= 2[xln|x-5| - 5ln|x-5| - x + 5 + C]

But 5 is a constant, so it gets embedded into C. Multiplying C by 2 still gives C.

= 2[xln|x-5| - 5ln|x-5| - x] + C