This question is a more specific question related to https://mathoverflow.net/questions/69900/asymptotics-for-the-number-of-ways-to-sum-primes-such-that-the-sum-is-n
I am looking for a lower bound (that is tight as possible) for the integral $\int_{2}^n e^{\sqrt{x/\log{x}}} dx.$ Unfortunatelly, $e^{\sqrt{x/\log{x}}}$ is not integrable, so one has to bound it first and then evalute the integral of the bounded function. For example $\int_{2}^n e^{\sqrt{x/\log{x}}} \geq \int_{2}^n e^\sqrt[3]{x} dx = O(n^{\frac{2}{3}}e^{\sqrt[3]{n}}) $
The bound obtained above is inferior to $e^{\sqrt{n/\log{n}}}$, so I should either find a better bound for the integrated function, or perhaps use another (unknown to me) trick to bound the stated integral.
Any suggestions are appreciated!