At our university we are now discussing changes to the course contents and there is some heated discussion regarding integration in the first year calculus courses. Currently, the techniques of exact integration include integration by parts, substitution, and lots of quite complicated formulas for the integrals of various trigonometric functions, and quite some emphasis on tricks for trigonometric substitutions, and integration by partial fractions.
I'm of the opinion that more emphasis should be placed on properly understanding integrals and only drill the most elementary techniques: by parts, substitution, partial fractions. I personally see little use for being able to compute complicated integrals by hand when a computer will do it in a split second. I see integration techniques disappearing from the standard tool-box of the mathematician, replaced by software. I thus see integration techniques as a niche and think that students should not be wasting much time on perfecting their integration skills. They just need to be able to solve simple integrals by hand.
I'm interested in hearing other opinions and, particularly from students, which integration techniques that you learned in your first year do you actually find useful in your later studies (whatever your discipline is).