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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
7 months ago