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Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from 2003 to 2012. Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. You may be able to find more up-to-date versions of some of these notes at http://www.cs.yale.edu/homes/aspnes/#classes.

First you have to know HowToDifferentiate. Having learned how to differentiate, your goal in integrating some function f(x) is to find another function F(x) such that F'(x) = f(x). You can then write that the integral of f(x) is F(x)+C (any constant C works), and compute definite integrals with the rule Integrala to b f(x) dx = F(b) - F(a).

How do you find this magic F(x)? Some possibilities:

• Memorize some standard integral formulas. Some useful ones for AlgorithmAnalysis are:

•  f(x) F(x) f(x)+g(x) F(x)+G(x) a f(x) [a constant] a F(x) f(ax) [a constant] F(ax)/a xn [n constant not equal to -1] xn+1/(n+1) x-1 ln x ex ex ax [a constant] ax/ln a [follows from ax = ex ln a] ln x x ln x - x
• Guess but verify. Guess F(x) and compute F'(x) to see if it's f(x). May be time-consuming unless you are good at guessing, and can put enough constants in F(x) to let you adjust F'(x) to equal f(x). Example: if f(x) = 2/x, you may remember the 1/x formula and try F(x) = a ln bx. Then F'(x) = ab/(bx) = a/x and you can set a = 2, quietly forget you ever put in b, and astound your friends (who also forgot the a f(x) rule) by announcing that the integral is 2 ln x. Sometimes if the answer comes out wrong you can see how to fudge F(x) to make it work: if for f(x) = ln x you guess F(x) = x ln x, then F'(x) = ln x + 1 and you can notice that you need to add a -x term (the integral of -1) to get rid of the 1.
• There's a complicated technique called "integration by parts" which is the integral version of the duv = u dv + v du formula, but it doesn't work as often as one might like. The rule is that Integral u dv = uv - Integral v du; an example is Integral ln x dx = x ln x - Integral x d(ln x) = x ln x - Integral x (1/x) dx = x ln x - Integral 1 dx = x ln x - x. You probably shouldn't bother memorizing this unless you need to pass AP Calculus again, although you can rederive it from the duv = u dv + v du formula.
• Use a computer algebra system like Mathematica, Maple, or Maxima. Mathematica's integration routine is available on-line at http://integrals.wolfram.com.

• Look your function up in a big book of integrals. This is actually less effective that using Mathematica, but may continue to work during power failures.

2014-06-17 11:58