Integration 25

We can simplify the integrand using identities $2 \sin A \sin B = \cos (A-B) - \cos (A+B)$ and $2 \cos A \cos B = \cos (A-B) + \cos (A+B)$ .

$\begin{aligned} I & = \int_0^{2\pi} \sin x \sin 2x \sin 3x \sin 4x\ dx \\ & = \int_0^{2\pi} \frac {(\cos x - \cos 3x) (\cos x - \cos 7x)}4 dx \\ & = \frac 14 \int_0^{2\pi} \left(\cos^2 x - \cos x \cos 3x - \cos x \cos 7 x + \cos 3x \cos 7x \right) dx \\ & = \frac 14 \int_0^{2\pi} \frac 12 \left(\left(1+\cos 2x\right) - \cos 2x - \cos 5x - \cos 6x - \cos 8 x + \cos 4x + \cos 10x \right) dx & \small \blue{\text{Note that }\int_0^{2\pi} \cos nx \ dx = 0} \\ & = \frac 18 \int_0^{2\pi} dx = \frac x8 \ \bigg|_0^{2\pi} = \frac \pi 4 \approx \boxed{0.785} & \small \blue{\text{where }n \text{ is an integer.}} \end{aligned}$