A designer is planning the curved profile of a long roof…

A designer is planning the curved profile of a long roof. She places the wall at $x = 1$ metre and measures $x$ along the ground in metres. The height of the roof above the ground is $y$ metres, and at the wall the height is $0$ metres, so $y = 0$ when $x = 1$. The slope of the roof at distance $x$ is $\dfrac{dy}{dx} = \dfrac{1}{x\left(1 + (\ln x)^2\right)}$ for $x \ge 1$. She sees that the factor $\dfrac{1}{x}$ is the derivative of $\ln x$, and that $1 + u^2$ in the bottom is a standard form. She wants the equation of the roof, and then the height at $x = e$ metres and at $x = e^2$ metres. She also wants to know how high the roof can get. As $x$ becomes very large, the height tends to a limit, and no point of the roof is above that limit. Find the heights at $x = e$ and at $x = e^2$, and the limit of the height. Give each height in metres to four decimal places.