Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$ How would you approach
$$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$
The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ways. Thanks !

Let $I=\displaystyle\int_0^{\frac\pi2} \frac{\sin (2n+1)x}{\sin x} dx$
As $\displaystyle\int_a^bf(x)dx=\int_a^bf(a+b-x)dx,$
$\displaystyle I=\int_0^{\frac\pi2} \frac{\sin (2n+1)(\frac\pi2-x)}{\sin (\frac\pi2-x)} dx$
$\displaystyle =\int_0^{\frac\pi2} \frac{\sin \{n\pi+\frac\pi2-(2n+1)x\}}{\cos x} dx$
$\displaystyle =\int_0^{\frac\pi2} \frac{\cos (2n+1)x}{\cos x} dx$ if $n$ is even.
$\displaystyle =-\int_0^{\frac\pi2} \frac{\cos (2n+1)x}{\cos x} dx$ if $n$ is odd.
If  $n$ is odd, 
$\displaystyle 2I=\int_0^{\frac\pi2} \frac{\sin (2n+1)x}{\sin x} dx-\int_0^{\frac\pi2} \frac{\cos (2n+1)x}{\cos x} dx$
$\displaystyle =\int_0^{\frac\pi2} \frac{\sin (2n)x}{\sin x\cos x} dx$
$\displaystyle =2\int_0^{\frac\pi2} \frac{\sin (2n)x}{\sin 2x} dx$
$\displaystyle =2\frac12\int_0^{\pi} \frac{\sin ny}{\sin y} dy$
$\displaystyle =2\int_0^{\frac{\pi}2} \frac{\sin ny}{\sin y} dy$ as $\displaystyle\frac{\sin ny}{\sin y}$ is an even function.
So, $\displaystyle\int_0^{\frac\pi2} \frac{\sin (2n+1)x}{\sin x} dx=I=\int_0^{\frac{\pi}2} \frac{\sin nx}{\sin x} dx$ if $n$ is odd.
Similarly, $\displaystyle \int_0^{\frac\pi2} \frac{\sin (2n+1)x}{\sin x} dx=\int_0^{\frac{\pi}2} \frac{\sin (n+1)x}{\sin x} dx$ if $n$ is even.
If we put, $2n+1=2013, n=1006$ which is even.
S0,  $\displaystyle \int_0^{\frac\pi2} \frac{\sin (2013)x}{\sin x} dx=\int_0^{\frac{\pi}2} \frac{\sin (1007)x}{\sin x} dx$
Now, if we put $2n+1=1007,n=503$ which is odd.
So, $\displaystyle \int_0^{\frac{\pi}2} \frac{\sin (1007)x}{\sin x} dx=\int_0^{\frac{\pi}2} \frac{\sin (503)x}{\sin x} dx$
Now, if $2n+1=503,n=251$
The reduction of $n$ will follow :$2013,1007,503,251,125,63,31,15,7,3,1$
So,$\displaystyle \int_0^{\frac\pi2} \frac{\sin (2013)x}{\sin x} dx=\int_0^{\frac{\pi}2} \frac{\sin x}{\sin x} dx=\frac{\pi}2$
