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Question
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Solution
Let I = `int cos 3x cos 2x cos x *dx`
Consider cos 3x cos 2x cos x = `(1)/(2) cos 3x [2 cos 2x cos x]`
= `(1)/(2)cos3x [cos(2x + x) + cos(2x - x)]`
= `(1)/(2)[cos^2 3x + cos3x cosx]`
= `(1)/(4)[2cos^2 3x + 2cos 3x cosx]`
= `(1)/(4)[1 + cos6x + cos(3x + x) + cos(3x - x)]`
= `(1)/(4)[1 + cos6x + cos4x + cos2x]`
∴ I = `(1)/(4) int[1 + cos6x + cos4x + cos2x]*dx`
= `(1)/(4) int 1*dx + 1/4 int cos6x*dx + 1/4 int cos4x*dx + 1/4 int cos2x*dx`
= `x/(4) + (1)/(4)((sin6x)/6) + 1/4((sin4x)/4) + 1/4((sin2x)/2) + c`
= `(1)/(48)[12x + 2sin 6x + 3sin 4x + 6sin2x] + c`.
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