How does differentiation under the integral sign

[autodidude]

I've read about it before and now I'm trying to learn it myself from Woods 'Advanced Calculus' (as well as other resources like http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf)

In the pdf, it says the method concerns integrals that depend on a parameter...now couldn't we make any function depend on a parameter?

For example inserting a parameter into $$f(x)=x^2$$ so it becomes $$f(x)=\alpha x^2$$

All the examples I've come across already have parameters in them so I'm not really sure. In 'Surely You're Joking, Mr Feynman!', Feynman says that he used this technique over and over again and often solved integrals that others couldn't using it. Would these integrals all be ones that depend on some parameter as well as a variable?

I'd like to know how exactly to use this method of integration and when to use it.

 

[mfb]

You can always add parameters to your functions. It is not useful everywhere, but you can do it.