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The value of the integral from 0 to pi/2 of [sin(8x)/sin(x)] dx is:

1. 152/105
2. 52/105
3. 52/35
4. 152/35

Correct answer: 152/105

Solution

Using the Chebyshev/product-to-sum identity, sin(8x)/sin(x) = 2cos(7x) + 2cos(5x) + 2cos(3x) + 2cos(x). Integrating each term from 0 to pi/2: integral of 2cos(kx)dx = 2sin(kx)/k. Evaluating at pi/2 and 0: [2sin(7pi/2)/7 + 2sin(5pi/2)/5 + 2sin(3pi/2)/3 + 2sin(pi/2)/1] - 0 = [2(-1)/7 + 2(1)/5 + 2(-1)/3 + 2(1)/1] = -2/7 + 2/5 - 2/3 + 2 = (-30 + 42 - 70 + 210)/105 = 152/105.

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