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Question
Integrate the following functions with respect to x :
`(sin4x)/sinx`
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Solution
`int (sin4x)/sinx "d"x = int (sin2(2x))/sinx * "d"x`
[sin 2A = 2 sin A cos A]
= `int (2sin 2x cos 2x)/sinx "d"x`
= `2 int (2sinx cosx * cos 2x)/sinx * "d"x`
= `4int cos 2x cos x * d"x`
= `2 int 2 cos 2x cosx * "d"x`
= `2 int [cos(2x + x) + cos(2x - x)] "d"x`
[2 cos A cos B = cos(A + B) + cos(A – B)]
= `2 int (cos 3x + cosx) "d"x`
= `2 int cos 3x "d"x + 2int cos x "d"x`
= `2 (sin3x)/3 + 2 sin x + "c"`
= `2 [(sin 3x)/3 + sin x] + "c"`
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