InterviewSolution
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InterviewSolution
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`int_(0)^(pi/4) (sec x)/(1+2 sin^(2)x)` is equal toA. `(1)/(3)[log(sqrt2+1)+(pi)/(2sqrt2)]`B. `(1)/(3)[log(sqrt2+1)-(pi)/(2sqrt2)]`C. `3[log(sqrt2+1)-(pi)/(2sqrt2)]`D. `3[log(sqrt2+1)+(pi)/(2sqrt2)]` |
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Answer» Correct Answer - A Let `I=int_(0)^(pi//4)(cosx)/(cos^(2)x(1+2sin^(2)x))dx` `=int_(0)^(pi//4)(cosxdx)/((1-sin^(2)x)(1+2sin^(2)x))` Out `t=sin x rArr dt = cos x dx` `therefore I=int_(0)^(1//sqrt2)(1)/((1-t^(2))(1+2t^(2)))dt` `=(1)/(3)int_(0)^(1//sqrt2)((1)/(1-t^(2))+(2)/(1+2t^(2)))dt` `=(1)/(3)[(1)/(2.1)log((1+t)/(1-t))+(2)/(sqrt2)tan^(-1)tsqrt2]_(0)^((1)/(sqrt2))` `=(1)/(3)[(1)/(2)log((sqrt2+1)/(sqrt2-1))+sqrt2tan^(-1)1]` `=(1)/(3)[(1)/(2)log(sqrt2+1)^(2)+sqrt2.(pi)/(4)]=(1)/(3)[log(sqrt2+1)+(pi)/(2sqrt2)]` |
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