1

I have to evaluate

$$\int_0^{\frac{\pi}{2}}\log(a^2\cos^2x+b^2\sin^2x)dx.$$

Now I have arrived at the answer by separating the original integral into integral $\log(a^2\cos^2x)$ plus integral of $\log(t^2\tan^2x)$, and then using differentiation through the integral sign on each of these. Then we can consider each of the a=0 or b=0 cases separately. I arrive at the correct answer of $\pi*\log((|a|+|b|)/2).$

However, it is more fiddly to check the conditions under which differentiating through the integral sign is satistfied. I need to check for $\log(a^2\cos^2x)$ and $\log(t^2\tan^2x)$ that: (i) These are integrable on 0 to $\pi/2$. (ii) Their derivative (which exists) is bounded by functions g1(x), g2(x) respectively (g1, g2 are independent of a and t respectively).

Any ideas on how I should tackle this?

Thanks.

Cameron L. Williams
  • 30,485
niswas1
  • 21
  • See http://math.stackexchange.com/questions/94628/differentiation-under-the-integral-sign-for-lebesgue-integrable-derivative. – Martín-Blas Pérez Pinilla Mar 17 '14 at 18:26
  • Thanks, but for this particular question how do I find show: (i) each of the integrands are integrable (without calculating them) and (ii) Find functions which uniformly bounds the derivatives? – niswas1 Mar 18 '14 at 00:49
  • Any updates guys? I think this may work: Note that we can show the required properties for $(a,b)$ for any $0<a<b<pi/2$. Hence this properties should also hold on $(0,pi/2)$. Please confirm if this is ok. – niswas1 Mar 20 '14 at 20:03

0 Answers0