Any hints for this one please?
$$\int_{-\pi/2}^{\pi/2}\frac{1}{1+2009^x}\frac{\sin(2010x)}{\sin(2010x)+\cos(2010x)}\,\mathrm{d}x $$
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Wish I could use $$\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$$? – lab bhattacharjee Mar 24 '14 at 13:42
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1I tried that but it's not helping very much.. – Superbus Mar 24 '14 at 13:47
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7The integrand function is NOT integrable in a neighbourhood of a point $x$ for which $\sin(2010 x)+\cos(2010 x)$ vanishes. – Jack D'Aurizio Mar 27 '14 at 18:04
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1What is the source of this problem? The use of years ($2009, 2010$) is highly suggestive that this is from a college math competition. I have checked Putnam 2008-2011; this does not appear there. – MT_ Mar 30 '14 at 02:18
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1Source: near the bottom of http://www.physicsforums.com/showpost.php?p=3433157&postcount=272 – user85798 Mar 31 '14 at 07:31
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3@JackD'Aurizio And aren't there a infinitely many of those points? It would seem this integral is "duly divergent"… – Geremia Apr 02 '14 at 02:18
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Given $n$, there exist a $(\tau,\rho)\in\mathbb{R}^2$ such that $f(x+\tau)-\rho$ is an odd function over $I=(-\pi/2,\pi/2)$. Assuming that $f(x)\in L^1(I)$ (this is not the case, anyway) we just have $\int_I f(x),dx = \pi\rho.$ – Jack D'Aurizio Apr 02 '14 at 11:27
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See the answer to this question. – Lucian Apr 06 '14 at 06:57
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@Lucian, Would you mind sharing your solution once you are done – lab bhattacharjee Apr 06 '14 at 13:18
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@labbhattacharjee: Are you saying that this integral cannot be solved by the same method as the one I linked to ? – Lucian Apr 06 '14 at 13:28
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@Lucian, Frankly speaking I could not, but I'm eager to learn an elementary solution of this problem – lab bhattacharjee Apr 06 '14 at 13:29
1 Answers
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To expand on the comment of @Jack D'Aurizio: when does $$ \sin \left( k x \right) + \cos \left( k x \right) = 0? $$
Examine the plot for the case when $k=1$.
Given integer $j$, the roots are $$ x_{-} = \frac{-\frac{\pi}{4} + 2j \pi}{k}, \qquad x_{+} = \frac{\frac{3\pi}{4} + 2j \pi}{k} $$
Over the domain, there $2010$ roots for $x_{-}$ and another $2010$ roots for $x_{+}$.
The figure below zooms in on one of these $4020$ singularities.
Answer
Integral is not defined.
dantopa
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