Interesting Integral

I am currently preparing for IITJEE 2017.After class, my teacher gave me a sheet with a few indefinite integral questions.We got most of the questions but we couldn't solve one question.Nobody could solve it, which is when I decided to discuss about this problem online.The question is,

Integrate: \intln(1+x2)1x2\frac{ln(1+x^{2})}{\sqrt{1-x^{2}}}dxdx

I substituted xx = sinθ\sin \theta but I finally got an expression like

\intln(1+sin2θ)\ln(1+\sin^{2} \theta)dθd \theta

Then I converted sin2θ\sin^{2} \theta to 1cos2θ1-\cos^{2} \theta and applied the formula a2b2a^{2}-b^{2} = (ab)(a+b)(a-b)(a+b)

At last I got , \intln(2cosθ)ln(\sqrt{2}-\cos \theta)dθd \theta ++ \intln(2+cosθ)ln(\sqrt{2}+\cos \theta)dθd \theta

After this I got stuck.I tried many methods but I couldn't get a perfect solution.

I usually go to classes in morning.So if anyone replies during that time I won't be able to respond.I come home by 8pm. After that I can reply.Please post the solution if anyone has got a different method than mine.

Note by Dhanvanth Balakrishnan
4 years, 5 months ago

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