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

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Answer 1 · 2015-06-05T21:34:21

$2int 1*ln(x-5)dx$

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 $C$ .

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

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