<s>[INST] Tricky integration/functions problem For $x>0$, let $f(x) = \displaystyle \int_1^x\frac{\ln t}{1+t}dt$. Find the function $f(x) + f(1/x)$ and show that $f(e) + f(1/e) = 1/2$.
Any help would be thoroughly appreciated.
 [/INST] Let $g(x)=f(x)+f(\frac{1}{x})$.  Then $ g^{\prime}(x)=\frac{\ln x}{1+x}+\frac{\ln\frac{1}{x}}{1+\frac{1}{x}}\cdot\left(\frac{-1}{x^2}\right)=\frac{\ln x}{1+x}+\frac{-\ln x}{x+1}\cdot\left(\frac{-x}{x^2}\right)=\frac{x\ln x+\ln x}{x(x+1)}=\frac{(\ln x)(x+1)}{x(x+1)}=\frac{\ln x}{x}$,
so $g(x)=\frac{1}{2}(\ln x)^2+C$, and $C=g(1)=0$.
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