So, here's an example:
$$\int \frac{x^2}{\sqrt{1 - x^2}}\, \mathrm dx$$
My math's teacher just say Add and Subtract 1 like this
$$\int \frac{1 + x^2 - 1}{\sqrt{1 - x^2}}\, \mathrm dx.$$
This is just one example. This thing is in almost half of the problems.
I get the next steps but what disturbs me is WHERE should one multiple/divide or add/subtract a number to make it easier. Of course there must be some concept behind it which I don't know so it will be a great help. I have seen this thing last year while studying trigonometry and this year while studying Calculus. If someone can explain how does this happen. That would be really helpful :)