Challenge!!!!!!!!!!!!!

Latex from AOPS,so you are from goiit?
It is of the form

sin θ.sin 2θ.sin3θ.sin4θ.sin5θ
club sinθ.sin5θ,sin2θ.sin4θ and try to proceed further.

with the substitution t = x - pi/6, the integrand becomes f(t) = sint sin2t sin 3t sin 4t sin 5t between the limits -pi/6 to pi/6.

f(t) is an odd function as f(-t) = -f(t)

since the limits are symmetric about zero, the value of the integral = 0