I need to obtain the steady state value of temperature, which is prescribed by the following equation. (Find limit as $t \to \infty$)
$$T-T_0 = \frac{4P}{\rho C \pi\sqrt{4a\pi}}\int_{t'=0}^{t'=t}\frac{dt'(t-t')^{-0.5}}{\sigma^2 + 8a(t-t')}\exp\left[-\frac{2(x-vt')^2 + y^2}{\sigma^2 + 8a(t-t')} - \frac{z^2}{4a(t-t')}\right]$$
I tried making the graph using integral calculator, and I found it for random values for the constants. But I am unable to proceed. I think a Laplace transform might be required, and then using Final Value Theorem, but it seems like way too much effort. How should I proceed ?