I'm trying to solve the following integral numerically
$\tilde{B}_{2}\left(\tilde{r}\right)=1-3\left(\frac{1}{\sigma}\right)^{3}\int _{1}^{\infty}\left(e^{-\left(\tilde{W}_{VdW}+\tilde{W}_{EDL}+\tilde{W}_{hyd}\right)}-1\right) \, \tilde{r}^{2}d\tilde{r}.$
The W's are the different interaction energies in my system. My only problem in the numerical integration is in small region near $\tilde{r}=0$ (cutoff length). This is because some of the W's has a huge negative value (-10^5) and the exponential term explodes. The different W's are:
$\tilde{W}_{VdW}=-\frac{A}{12k_{B}T} \left[\frac{1}{\tilde{r}^{2}-1} +\frac{1}{\tilde{r}^{2}} +2\ln \left(1-\frac{1}{\tilde{r}^{2}}\right)\right],$ $\tilde{W}_{EDL}(\tilde{h})=-\frac{\pi R\beta}{k_{B}T\kappa^{2}}\left\{\frac{\sigma}{\kappa^{-1}}\left(\tilde{r}-1\right)-ln\left[2sinh\left(\frac{\sigma}{\kappa^{-1}}\left(\tilde{r}-1\right)\right)\right]+ln\left[tanh\left(\frac{\sigma}{2\kappa^{-1}}\left(\tilde{r}-1\right)\right)\right]\right\},$
$\tilde{W}_{hyd}(\tilde{r})=-\lambda e^{-\frac{\sigma}{D_{H}}\left(\tilde{r}-1\right)}.$
$\tilde{r}$ is the independent variable where all the rest are parameters. You can see that VDW and EDL are negative (attraction forces) therefore it turns the exponential term inside the integral to with positive argument - a big one! Becuase $\lambda$ and other parameters are relatively big (10^5). I did some asymptotic analysis to all three W's in small $\tilde{r}$'s in this small region and split the integral to two parts but the exp. term still explodes because I still have the singularity near $\tilde{r}=0$.
I will be happy to get some help from you guys. Much appreciation, Roi