Your best mathematical secrets... - Page 2

theodds · #16 $Old$ November 10th, 2009, 11:04 PM

Quote:

Originally Posted by artvandalay11 $View Post$

I wouldn't say the following makes me good at calculus but it definitely makes me much faster....

I try to avoid u-substitution at all costs when integrating. By this I mean I almost never actually write down let u= and then subsitute it back in so that I can see an obvious integration pattern.

I firmly believe that when learning calculus you should utilize and guess and test method in your head, and not write down u-subs. I've been integrating this way from the beginning, due to a fantastic teacher, and at the University level I was finishing my calc tests 25 minutes into the class. My professors were shocked and one even asked me where my u-sub was. I replied I don't need it, cuz there's your answer

Example: $\int \cos (5x)dx$

You should know that the answer is going to involve $\sin$
Moreover, it's going to involve $\sin (5x)$ because there's no other way to get the 5 in there

Now take the derivative in your head, you'll realize that you'll be off by a factor of 5, so stick a $\frac{1}{5}$ in front of your initial guess.

Obviously, I'm just not writing down the u-sub, but that's because you simply don't need to

At the same time, for some problems I would say you really need to u-sub or you're sunk. Obviously for things like 1/(3x +5), or whatever, you can do it in your head easily, but just as an example, today I had a problem that required two different, and IMO very non-obvious u-subs to get the desired result, which itself was an intractable integral (you could do it in one, but man....you would need to be Gauss to have the intuition to get figure out whatever convoluted sub that would have been). I know you're not saying "don't learn u-subs," and I think your general approach probably is more useful is developing that X-factor that makes a good mathematician.