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Integration using U-Substitution of Indefinite Integrals.

Ilustrations of several u-substitution problems

My lecture on substitution from Calc I

Slides from my Calc I Lecture

Here is another good video by Khan Academy explaining integration by substitution for those of you who may need help. There are also other videos on pretty much any topic in calculus which I find useful for reviewing and practicing. Khan Academy is on a mission to provide a free world-class education to anyone anywhere. With over 2,600 videos covering everything from arithmetic to physics, finance, and history and 200 practice exercises, we're helping students learn whatever they want, whenever they want, at their own pace.

Students have a lot of problems with changing the limits when doing u-substitution with definite integrals. But it's easy to remember: if the variable integrated over is u, the limits have to be values of u. You start with limits which are values of x. How do you convert x to u? We have already decided that u=cos(x). So the limits x=π/2 and x=π become cos(π/2) = 0 and cos(π) = -1. He's being a bit sloppy by saying this computes a negative area. Area is always positive. The integral *results* in a negative number because the region we are finding the area of is below the x-axis. So the integral computes the negative of the area.