$ \int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$ and $ \int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$

Asked Mar 15 '15 at 05:19

Active Mar 20 '15 at 12:33

Viewed 187 times

I tried solving this integral:

$$\int^{\pi/2}_{-\pi/2} \frac{1}{2007^x + 1}\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$

I took a while before aptly applying the following identity I had noted down in my notebook before.. $$\int^m_{-m}\phi(x)dx=\int^m_0\phi(x)+\phi(-x) dx$$

And my hunch was right; I managed to get it to a 'manageable' form: $$I_1=\int^{\pi/2}_0\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$

This got me wondering..

Does there exist a closed form for the following integrals?

$$ I_2=\int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$$$$ I_3=\int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$$

edited Mar 20 '15 at 12:33

Harry Peter

7,819

asked Mar 15 '15 at 05:19

Panglossian Oporopolist

5,223

@labbhattacharjee I've changed my question. – Panglossian Oporopolist Mar 15 '15 at 05:27
@Kugelblitz, Once marked duplicate, I think, better ask for an independent question instead of changing – lab bhattacharjee Mar 15 '15 at 05:30
@labbhattacharjee Please remove the duplicate sign; I've changed the question. – Panglossian Oporopolist Mar 15 '15 at 05:30
But why? @labbhattacharjee I was just following your suggestion; since you redirected me to the duplicate.. Besides, those two integrals were also part of my question; you can check it in the edit yourself!!!! – Panglossian Oporopolist Mar 15 '15 at 05:31
@labbhattacharjee Thank you sir; I will try harder to prevent duplicates next time! – Panglossian Oporopolist Mar 15 '15 at 05:32
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Related : http://math.stackexchange.com/questions/741580/integral-int-pi-2-pi-2-frac12007x1-cdot-frac-sin2008x-si – lab bhattacharjee Mar 15 '15 at 05:32
@labbhattacharjee Ahh... So integrals $I_2$ and $I_3$ are equal? – Panglossian Oporopolist Mar 15 '15 at 05:40
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Here is a solution for your problem. – ASB Mar 15 '15 at 05:47
@ARUNA No, that's the solution for the problem I was working on; I need help in evaluating $I_2$ and $I_3$ – Panglossian Oporopolist Mar 15 '15 at 06:27
@Kugelblitz, Are $I_2,I_3$ indefinite versions of some definite ones? – lab bhattacharjee Mar 15 '15 at 08:15
@labbhattacharjee As far as I know, no. I just thought of these generalized integrals after having tried to solve the question you initially marked duplicate. – Panglossian Oporopolist Mar 15 '15 at 08:51

$ \int \frac{\sin^{a}x}{\sin^{a}x+\cos^{a}x}dx$ and $ \int \frac{\cos^{a}x}{\sin^{a}x+\cos^{a}x}dx$

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