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For`x >0,l e tf(x)=int_1^x((log)_e t)/(1+t)dtdot`Findthe function `f(x)+f(1/x)`andshow that `f(e)+f(1/e)=1/2dot` |
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Answer» Correct Answer - `[(1)/(2)(Inx)^(2)]` f(x) `=int_(1)^(x)(Int)/( 1+t)dt " for" x gt 0` Now , `f(1//x)=int_(1)^(1//x)(Int)/( 1+t)dt` Put `t=1 //u rArr dt = (-1//u^(2))du` `:. f (1//x)= int_(1)^(x)(In(1//u))/(1+1//u)*((-1))/(u^(2))du` Now , `f(x) + f((1)/(x))= int_(1)^(x) (Int)/((1 +t))dt + int_(1)^(x) (Int)/(t(1+t))dt` `=int_(1)^(x)((1+t)Int)/(t( 1+t))dt = int_(1)^(x)(Int)/(t) dt = (1)/(2)[(In t)^(2)]_(1)^(x)=(1)/(2) (In x)^(2)` Put x = e , `f(e) + f ((1)/(e))=(1)/(2)(In e)^(2)=(1)/(2)` Hence proved. |
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