In Stewart's Calculus (7e) it is considered an improper integral when the interval is infinite or when the f has an infinite discontinuity at [a,b].

I guess you should just remember to replace the number(or infinity)  where the discontinuity occurs by a letter (for example t) and remember where the sin, cos, and tan is discontinuous...After that just use the previous techniques like l'hopitals, chain rule, etc. 

There are a lot of sample long tests online and you can also check patrickjmt's videos on youtube:)