Finding value of $\lim _{n\to\infty} \int_0^\pi \frac{\sin x}{1+\cos^2(nx)}dx$

Asked May 23 '20 at 12:30

Active May 23 '20 at 12:39

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We have to calculate $$\lim _{n\to\infty} \int_0^\pi \frac{\sin x}{1+\cos^2(nx)}dx$$ Now I thought that perhaps the integral needs to be calculated in terms of $n$ and then we plug in the limit but I could find no way to do it.

Perhaps it is non-integrable as an indefinite integral but I couldn't find any way to do this even as a definite integral. I observed the fact that the integrand is an even function but even that just serves to reset the limits of the integral.

So now I'm completely stuck. I think Sandwich theorem may be used by establishing two estimates for the integral which are close to the integrand but I'm not sure how to do this. Any help would be appreciated.

edited May 23 '20 at 12:39

asked May 23 '20 at 12:30

infinite-blank-

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I don't see why the limit even exists... – David C. Ullrich May 23 '20 at 12:39
@DavidC.Ullrich Can you please elaborate? – infinite-blank- May 23 '20 at 12:40
??? I don't know how to "elaborate" on that - I don't see why the limit exists. – David C. Ullrich May 23 '20 at 12:42
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If somehow proven not to exist this would be the answer, IMO. – Alexey Burdin May 23 '20 at 12:44
@DavidC.Ullrich: the limit exists as shown in the original (of which this is duplicate). – Paramanand Singh May 23 '20 at 12:49
Another answers are here and here – Alexey Burdin May 23 '20 at 12:58

Finding value of $\lim _{n\to\infty} \int_0^\pi \frac{\sin x}{1+\cos^2(nx)}dx$

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