Let $X$ be a random variable with the following pdf:
$$\displaystyle f(x) = \alpha \cdot e^{-x^2- \beta x}, x \in \mathbb{R} .$$
Also, $E[X]=-0.5$ . Well the problem is straightforward except that I am not able to integrate this function. How can I evaluate this integral?
Alternative approach rather than integration:
Recall that the pdf for normal distribution when $\mu=-\frac12$ is
\begin{align}\frac1{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2} \right)&=\frac{1}{\sqrt{2\pi\sigma^2}}\exp \left(-\frac{(x^2-x+\frac14)}{2\sigma^2} \right)\\ &= \frac{1}{\sqrt{2\pi\sigma^2}}\exp \left(-\frac{1}{8\sigma^2} \right)\exp \left(\frac{-x^2+x}{2\sigma^2} \right)\\\end{align}
Hence $2\sigma^2=1$, and hence $\alpha=\frac{1}{\sqrt{\pi}}\exp\left( -\frac14\right)$ and $\beta=-1$.