How to plot y(x) v.s x when y(x) is the upper limit of integration?

Question

I have a funciton in the following form. But I don't know how to plot it.

$$x=\int_1^{y(x)}\frac{dt}{\sqrt{2(t-t\ln t)}}.$$

For example, I want to plot it when $0<x<3$. Thanks in advance.

Answer 1

We must ask ourselves when does the integral exist? We see that when $\ln{t}>1$, we have the square root of negative number, which does not really make sense when we are talking about real variables. So, we say the integral maps y onto x when $0<y<e$. Outside of this range either the natural log or the square root fails to exist, so there is nothing to plot.

If we tell Mathematica about the range of $y$, we can get a symbolic expression.

h = Integrate[1/Surd[2 t (1 - Log[t]), 2], {t, 1, y}, 
   Assumptions -> 1 < y < E];
xmax = h /. y -> E
soln = Solve[x == h, y] // First;
Plot[y /. soln, {x, 0, xmax}]

The temptation is to go ahead and plot the "function" y/.soln on the interval $1<x<3$, but that would be wrong. We note that Solve gives us a warning that inverse functions are being used. What has happened is the inverse function introduced an extraneous root. When we evaluate NSolve[3 == h, y] we find that there is no upper limit $y$ for which our expression h is 3.