InterviewSolution
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InterviewSolution
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Evaluate `int(cos x)/((1-sinx)(2-sinx))dx.` |
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Answer» Putting sin `x=t` cos `x dx =dt,`we get `I=(cosx)/((1-sinx)(2-sinx))dx=int(dt)/((1-t)(2-t)).` `Let (1)/((1-t)(2-t))=(A)/((1-t))+(B)/((2-t))` `implies 1-=A(2-t)+B(1-t).` Putting `t=1` in (i) , we get `A=1. . . . (i).` Putting `t=2`in (i) , we get `B=-1.` `therefore (1)/((1-t)(2-t))=(1)/((1-t))-(1)/((2-t))` `implies int (cosx)/((-sinx)(2-sinx))dx=int(dt)/((1-t)(2-t))` `=int {(1)/((1-t))-(1)/((2-t))}dt` `=int (dt)/((1-t))-int(dt)/((2-t))` `=-log|1-t|+log|2-t|+C` `=log |(2-t)/(1-t)|+C=log |(2-sinx)/(1-sinx)|+C.` |
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