[SOLVED] Integral formulas

Hi MHF!

I need your help for some questions.

First, what is the method to find out that $\displaystyle \int \frac{dx}{1+x^2}=arctan(x)+C$ ? I find it stupid to learn it by heart. There are plenty of other formulas to know, so I would like to know how to recover them without learning all of them.

Second, I have to find $\displaystyle \int \frac{dx}{\sqrt{x^2-9}}$ in term of trigonometric or hyperbolic functions. Looking at my book, I found out that it is equal to $\displaystyle ln|x+\sqrt{x^2-9}|+C$. Which recalls me of an inverse of $\displaystyle cosh^{-1}$ but cannot find exactly to what it's equal, so I'd appreciate help for this one.

And finally $\displaystyle \int \frac{dx}{\sqrt{4-x^2}}=arcsin\big(\frac{x}{2}\big)+C$ according to my book. But once again, how can I know it's worth an inverse of the sine function? I hope I don't have to learn this by heart.