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Question
Find the integrals of the function:
sin x sin 2x sin 3x
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Solution
Let `I = int sin x sin 2x sin 3x dx`
`= 1/2 int (2 sin x sin 2x) sin 3x dx`
`= 1/2 int (cos x - cos 3x) sin 3x dx` ... [∵ 2 sin A sin B = cos (A - B) - cos (A + B)]
`= 1/4 int 2 sin 3x cos x dx - 1/4 int 2 sin 3x cos 3x dx` .... [∵ 2 sin A cos B = sin (A + B) + sin (A - B)]
`= 1/4 int (sin 4x + sin 2x) dx - 1/4 int sin 6x dx`
`= -1/16 cos 4x - 1/8 cos 2x + 1/24 cos 6x + C`
`= 1/4 [1/6 cos 6x - 1/4 cos 4x - 1/2 cos 2x] + C`
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