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The integral `int_(1)^(e){((x)/(e))^(2x)-((e)/(x))^(x)} "log"_(e)x` dx is equal toA. `(3)/(2)-e-(1)/(2e^(2))`B. `-(1)/(2)+(1)/(e)-(1)/(2e^(2))`C. `(1)/(2)-e-(1)/(e^(2))`D. `(3)/(2)-(1)/(e)-(1)/(2e^(2))` |
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Answer» Correct Answer - A Let `I=int_(1)^(e){((x)/(e))^(2x)-((e)/x)^(x)} "log"_(e) x dx` Now , put `((x)/(e))^(x)=t rArrx log _(e)((x)/(e))= log t` `rArrx (log_(e)x-log_(e))=logt` `rArr[x((1)/(x))+(log_(e)x-log_(e)e)]dx=(1)/(t)dt` `rArr(1+log_(e)x-1)dx=(1)/(t)dtrArr(log_(e)x)dx=(1)/(t)dt` Also , upper limit `x=erArrt=1` and lower limit `x=1rArrt=(1)/(e)` `I=int_(1//e)^(1)(t^(2)-(1)/(t))*(1)/(t)dtrArrI=int_(1//e)^(1)(t-t^(2)) dt` `I=[((t^(2))/(2)+(1)/(t))]_(1/(e))^(1)={((1)/(2)+1)-((1)/(2e^(2))+e)}=(3)/(2)-e-(1)/(2e^(2))` |
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