Evaluating some Log Trig integrals through the derivatives of Beta Function
In this previous blog entry we have defined the beta function and derived some integral representations. We also mentioned that it´s very useful to compute a certain class of integrals. And this is the aim of today´s post. Lets start from the following representation (equation (8) in this post ) (1) Let´s differentiate the R.H.S of (1) with respect to Now recall the definition of the digamma function rearranging therefore (2) Lets now compute rewrite (2) as differentiating w.t. substituting (2) in the above equation (3) Similarly we can compute Differentiating (2) w.r. to Using (2) we get (4) On the other hand we may compute the derivatives of the L.H.S of (1) (5) (6) (7) (8) (9) Note that if we set and in (5), (6), (7), (8) and (9) we get (10) (11) (12) (13) (14) In order to evaluate (10), (11), (12), (13) and (14) we have to find the values of , ...