An intuitive understand of integration by parts is more valuable for a math student than this algorithm. (Although I doubt most high school calculus students could explain integration by parts a year later.)
Symbolic integration has a "problem solving" feel to it in high school and teachers generally focus on this part -- making many students feel stupid in the process if they can't figure it out.
Why not just tell them there's an algorithm for it and allow them to use a program to do the symbolic integration whenever necessary while solving an applied problem?
I would say that learning "problem solving" is more important than learning how to type a formula into integrals.wolfram.com (which you should certainly use if you have an actual applied problem). Plus, if you want to go on to multivariable calculus you will need to understand the principles rather than just being able to get an answer to an integration.
I agree that calculus seems too "magical". But we should solve that by explaining more of the intuition and heuristics behind solving calculus problems, rather than teaching people to solve problems by brute force.
Guys, stop downvoting posts just because you disagree with them. If you have something to add to the discussion, add it. Parent was not abusive or off-topic or otherwise deserving of downmodding.
Symbolic integration has a "problem solving" feel to it in high school and teachers generally focus on this part -- making many students feel stupid in the process if they can't figure it out.
Why not just tell them there's an algorithm for it and allow them to use a program to do the symbolic integration whenever necessary while solving an applied problem?