"Nasser M. Abbasi" schrieb: > > On a related point, would you please help me understand how > 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).