The question states to prove beta= sqrt(pi ln 5) whereas I get beta= pi ln 5. I attached my steps below
You have mishandled the limits of integration when making the substitution in the integral. If $\displaystyle t=\theta^2$ then $\displaystyle t=0$ when $\displaystyle \theta=0,$ and $\displaystyle t=\beta^2$ when $\displaystyle \theta=\beta$. So the integral should become $\displaystyle \displaystyle\int_0^{\beta^2}\!\!\!\tfrac12\sqrt t e^{2t/x}\tfrac1{2\sqrt t}}\,dt.$
You have worked out the integral correctly (apart from having $\displaystyle \sqrt t$ instead of $\displaystyle \beta^2$ as the upper limit of integration), and it leads to the result $\displaystyle 5 = e^{\beta^2/\pi}$, which is what you want.