Integral question
08-12-2017, 07:49 PM
Post: #1
 lrdheat Senior Member Posts: 574 Joined: Feb 2014
Integral question
Is there any way to get ((pi)^2)/4 for the integral from 0 to pi of (x*(sin x))/(1+ (cos x)^2) ?
08-13-2017, 12:15 AM (This post was last modified: 08-13-2017 12:44 AM by Helge Gabert.)
Post: #2
 Helge Gabert Senior Member Posts: 467 Joined: Dec 2013
RE: Integral question
Yes, it can be done.

Note that arctan(1)-arctan(0) is being recognized as the desired symbolic solution (pi^2/4).

Just kidding. The hard part is to get there through some clever symmetrical substitution like u=pi-x, and some other substitutions, a shown here

https://artofproblemsolving.com/communit...x__on_0_pi

Not sure if that recognition pattern has been implemented in Giac/Xcas (maybe it is too expensive).
08-13-2017, 01:00 AM
Post: #3
 Mark Hardman Senior Member Posts: 522 Joined: Dec 2013
RE: Integral question
(08-12-2017 07:49 PM)lrdheat Wrote:  Is there any way to get ((pi)^2)/4 for the integral from 0 to pi of (x*(sin x))/(1+ (cos x)^2) ?

ibpu((x*sin(x)/(1+cos(x)^2)),x,x,0,π)

Ceci n'est pas une signature.
08-13-2017, 01:24 AM
Post: #4
 Helge Gabert Senior Member Posts: 467 Joined: Dec 2013
RE: Integral question
Excellent! Didn't think about ibpu and ibpdv. That'll do it.
08-13-2017, 03:24 PM
Post: #5
 lrdheat Senior Member Posts: 574 Joined: Feb 2014
RE: Integral question
Thanks!
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