I'm reading through the solutions of my book and they skip over a lot of steps so can someone please explain where their answers came from? I haven't done probability in a while so if there are some basic principles or theorems I'm forgetting I wouldn't be surprised.
1) I'm given that $X$ is a standard normal RV. The book says $E[e^{uX+vX}] = e ^{(u+v)^2/2}$. Where did that come from? I thought that I would need to do $\int_{-\infty}^{\infty}e^{-.5x^2}e^{(u+v)x}dx$, which gives a different answer.
2) Not expected value per se, but I have $\int_{x-a/2}^{x+a/2}e^{-.5x^2}dx$ and again the book just jumps to the answer that its $(a)e^{-x^2/2}$. Again, how did they get there?
For 1), looking at how to compute the MGF of the standard normal distribution may help. Note that your integral is not quite right (you forgot the constant $(2\pi)^{-1/2}$).
For 2) the integral does not make sense because a) you are using $x$ for two different meanings. Once you fix this, the "answer" is not exact, but an approximation using the fundamental theorem of calculus.