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Solution: Let I = ∫ 3 + 2 cos x / (2+ 3 cos x)2 dx
Multiplying Nr. & Dr. by cosec2x
=> I = int {{(3cosec^{2}x+ 2cot x cosec x) } / (2cosec x +3cot x)^2 }dx = - int {{(-3cosec^{2}x- 2cot x cosec x) } / (2cosec x +3cot x)^2 }dx
Let us consider 2cosec x +3cot x=t
then dt = (-3cosec^{2}x- 2cot x cosec x).dx
=int (dt / t^2)
integrate it then we get ={1} / {(2cosec x +3cot x)} ={sin x} / {2+3cos x} +c
given ∫ {(2cosx+3)/(2+3cosx)^2} dx
I =∫{(2cosx +3)/(2+3cosx)^2} dx
let multiplying N.R and D.R with cosec^2 x
then we get
I = - ∫ {(-2cosec x.cotx - 3cosec^2 x)/(2cosecx+3cotx)^2}.dx
let us consider t=2cosecx+3cotx
then dt =( -2cosec x.cotx - 3cosec^2 x ).dx
= -∫-(1/t^2) => 1/t => 1/(2cosecx+3cotx.cotx)
={(sinx) /(2+3cosx)}
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