TITLE: Evaluate Integral (Romanian Olympiad)
QUESTION [12 upvotes]: $$ \int\cos x\cdot\cos^2(2x)\cdot\cos^3(3x)\cdot\cos^4(4x)\cdot\ldots\cdot\cos^{2002}(2002x)dx $$
Taken from the 2002 Romanian olympiad

REPLY [3 votes]: If this is taken over $(0,2\pi)$ (or any multiple of this period):
By symmetry:
$$\int_0^{2\pi}\prod_1^{n}\cos^k kx\,dx=2\int_0^{\pi}\prod_1^{n}\cos^k kx\,dx$$
Now let $x\to \dfrac{\pi}{2}-x$ and denote the integral like so:
$$\int_{-\frac{\pi}{2}}^{ \frac{\pi}{2}}f(x)\,dx$$
Whenever $k$ is odd the transformation gives $\cos^k kx\to \pm\sin^k kx$ and $\cos^k kx\to \pm\cos^k kx$ otherwise. Hence $f$ is odd if there are an odd number of sines in the resulting product i.e. if there is an odd amount of odd numbers between $1$ and $n$.
Since there are exactly $1001$ odd $k$ such that $1\leq k\leq 2002$, the integrand is odd and therefore:
$$\int_0^{2\pi}\prod_1^{2002}\cos^k kx\,dx=0$$