The antiderivative of f(x)=log⁡(log⁡x)+(log⁡x)−2 whose graph passes through (e,e) is

An antiderivative of f (x) = F(x) =∫log⁡(log⁡x)+(log⁡x)−2dx+C=xlog⁡(log⁡x)−∫xxlog⁡xdx+∫(log⁡x)−2dx+C (integrating by parts the first term) =xlog⁡(log⁡x)−x(log⁡x)−1+∫(log⁡x)−2dx+∫(log⁡x)−2dx+C (again integrating by parts (log⁡x)−1 =xlog⁡(log⁡x)−x(log⁡x)−1+CPutting x=e we have e=0−e+C so C=2eThus F(x)=xlog⁡(log⁡x)−(log⁡x)−1+2e