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Topic: The Charlwood Fifty
Replies: 52   Last Post: Jun 24, 2013 10:24 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The Charlwood Fifty
Posted: Jun 17, 2013 6:09 PM

"Nasser M. Abbasi" schrieb:
>
> free version of reduce transformed
>
> arcsin(x)*log(x)
>
> to
>
> arcsin( sin(g0) ) * cos(g0) * log( sin(g0) )
>
> by replacing x with sin(g0).
>
> i.e Where does cos(g0) term come from in the above transformation?
>
> Here is a link to the reduce trace for integral #1, which it
> could not do btw. And the above was the first step in the process.
>
> http://12000.org/my_notes/ten_hard_integrals/reduce_logs/1/HTML/trace_1.html
>

In the original INT(ASIN(x)*LN(x), x), Reduce makes the variable
substitution x = SIN(y), dx = COS(y)*dy (I've written y for g0). So the
factor COS(y) just represents the derivative dx/dy that must be included
in the transformed integrand.

It looks like the integration process is simply restarted after the
variable substitution. While the first step is easy to understand, I
have no clear idea what Reduce is trying to do later - and without
success. It may be the Risch-Norman heuristic again.

By the way, Derive 6.10 attacks the original integral using integration
by parts. It can also evaluate INT(x*COS(x)*LN(SIN(x)), x) which is
obtained when ASIN(SIN(x)) in the variable-substituted integral is
replaced by x (this replacement is not valid for all complex x, nor for
all real x, however).

Martin.