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