Let it first be said that the kudos goes to CaptainBlack (but I'll hold it in safe-keeping - for a while

)

It works for integrands of the form:

(an exmaple has already been seen).

or

,

where you would observe that there exists polynomials

of degree n such that

gives the integrand.

Try integrating

this way. There's a subtle difference to the example just done ....

If you think about it a bit, you'll think of more.

The idea boils down to making a shrewd guess or observation as to the form of function which, when differentiated, gives the integrand. The technique is sometimes refered to as

*integration by recognition*.