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Evaluating indefinite integrals

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This article describes a number of methods for working out integrals of the form \int_a^bf(x)dx, where a and b are variables rather than constants. This amounts to finding a function F that differentiates to f, in which case \int_a^bf(x)dx=F(b)-F(a) (by the fundamental theorem of calculus).

When written, it should contain accounts of integration by parts and substitution, and also the trick of just spotting that the integrand has the form f(g(x))g'(x) (in which case you find a function F that differentiates to f and your answer is F(g(x))). Another trick is guessing a function F and then adjusting it afterwards. (For example, to integrate \log x, you could guess x\log x, since that will at least give you a \log x term in the answer, and then subtracts x to get rid of the extra 1 that you don't want.) Most of the article should be understandable by bright high-school students.