For `x >0,l e tf(x)=int_1^x(logt)/(1+t)dtdot` Find the function `f(x)+f(1/x)` and find the value of `f(e)+f(1/e)dot`

Explanation

Let `F(x)=(x)+f(1/x)`
`=int_(1)^(x)(logt)/(1+t)dt+int_(1)^(1//x)(logt)/(1+t)dt`
In second integral let `t=1//y`. So `dt=-1/(y^(2))dy`
`:.F(x)=int_(1)^(x)(logt)/(1+t)dt+int_(1)^(x)(-logy)/(1+1/y)(-(dy)/(y^(2)))`
`=int_(1)^(x)(logt)/(1+t)dt+int_(1)^(x)(logy)/(y(1+y))dy`
`=int_(1)^(x)(logt)/(1+t)dt+int_(1)^(x)(logt)/(t(1+t))dt`
`=int_(1)^(x)(logt)/t dt=1/2(logx)^(2)`
`:.F(e)=1/2`