# Evaluate the integral $\int_{-\pi/2}^{\pi/2}\frac{1}{1+2009^x}\frac{\sin(2010x)}{\sin(2010x)+\cos(2010x)}\,\mathrm{d}x$

Any hints for this one please?

$$\int_{-\pi/2}^{\pi/2}\frac{1}{1+2009^x}\frac{\sin(2010x)}{\sin(2010x)+\cos(2010x)}\,\mathrm{d}x$$

• I tried that but it's not helping very much.. – Superbus Mar 24 '14 at 13:47
• The 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
• What 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. – MCT Mar 30 '14 at 2:18
• Source: near the bottom of physicsforums.com/showpost.php?p=3433157&postcount=272 – user85798 Mar 31 '14 at 7:31
• @JackD'Aurizio And aren't there a infinitely many of those points? It would seem this integral is "duly divergent"… – Geremia Apr 2 '14 at 2:18

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.