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Integrate: (3 +2cosx)/(2+3cosx)^2

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Solution: Let ,

              I=int 3+2cosx/(2+3cosx)^2hspace0.1cmdx

          Divide above and below by sin^2x , then

            I=int (3cosec^2x+2cosecxcdot cotx)/(2cosecx+3cotx)^2hspace0.1cmdx

    put ,    (2cosecx+3cotx)=t   Rightarrow (-2cosecxcdot cotx-3cosec^2x)dx=dt

            \Rightarrow I=-int dt/t^2=1/t+c=1/(2cosecx+3cotx)hspace0.1cm+c\ \Rightarrow I=sinx/(2+3cosx)+c

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Deependra Verma

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