 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-2/section-a/2/a/solution

The product-to-sum identity for the sine gives

sin 2 x sin 4 x = \frac{1}{2} (cos 2 x - cos 6 x)

. Each cosine has zero integral on this interval, so

I = 0.

This is also an instance of Fourier orthogonality.

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