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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: \(\int\)\(\frac{ln(1+x^{2})}{\sqrt{1-x^{2}}}\)\(dx\)

I substituted \(x\) = \(\sin \theta\) but I finally got an expression like

\(\int\)\(\ln(1+\sin^{2} \theta)\)\(d \theta\)

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

At last I got , \(\int\)\(ln(\sqrt{2}-\cos \theta)\)\(d \theta\) \(+\) \(\int\)\(ln(\sqrt{2}+\cos \theta)\)\(d \theta\)

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
10 months ago

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