Why evaluate integrals when computers can do it?
Computers are pretty good at doing grubby and tedious chores - people are not. It would be nice if we could concentrate on the high-level stuff and let the computer handle the details. Things are never so neat though. Understanding is reluctant to visit us and frequently requires hard, grubby, detail work first. You practice doing integrals so you can learn how things work and so you can understand arguments based on integration. It's true that the details of integrating some combination of sines and cosines will not stay with you and will probably not be important to you - after all the computer can do it - but the pain and sweat will give you insight and understanding that you are unlikely to come by any other way. The basic ideas and the major techniques - substitution, integration by parts, partial fractions - are important, should be mastered, and will be mastered through hard work, not by punching a keyboard.
Old Fogey - Or you didn't buy it?
You didn't buy the comment above, did you? You think if I had my way, you'd be using a slate and chalk for all your work, a pointed stick and sand, or worse. Well maybe so. As we grow older our brains ossify and new methods seem immoral, or at least not pedagogically sound. Alright, if you are not going to learn how to evaluate integrals at least learn how to use a first-rate tool like Maple or Mathematica effectively. Do also learn how to apply Calculus effectively and learn thoroughly the main theorems and ideas, some of which are the chain rule, solution of extremal problems, the Fundamental Theorem, the intermediate value theorem, the mean value theorem, change of variable in an integral, and, yes, integration by parts.