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sin2x can be written as sin(5x-3x)=sin5xcos3x-sin3xcos5x
sin2x/sin5x.sin3x=sin5xcos3x-sin3xcos5x/sin5xsin3x
=cot3x-cot5x
integral of cot3x-cot5x=(1/3) logsin3x - (1/5) logsin5x+ c
Hope it helps
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