Iit Jee Integral 3+2Cosx/(2+3Cosx)^2, Integrate: (3 +2Cosx)/(2+3Cosx)^2



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Đang xem: Integral 3+2cosx/(2+3cosx)^2


Solution:Let I =∫ 3 + 2 cos x / (2+ 3 cos x)2dx

MultiplyingNr. & Dr. bycosec2x

=> 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



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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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