SUNY Geneseo Department of Mathematics
Math 221 02
Fall 2020
Prof. Doug Baldwin
(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)
The substitution method is a powerful tool for finding antiderivatives (i.e., indefinite integrals), so it’s not surprising that it’s also useful for finding definite integrals. Applying substitution to definite integrals involves applying the substitution to the bounds of integration as well as to the integrand, as discussed in the “Substitution for Definite Integrals” part of section 5.5 of our textbook. The following problems give you a chance to work with and practice this idea. See what you can do with the following integrals, what questions you have, etc....
\[\int_1^2 2x\sqrt{x^2-1}\,dx\] \[\int_0^\pi \cos(2\Theta)\,d\Theta\]And here are a couple more examples for more practice...
\[\int_0^{\frac{\pi}{8}} \sin^2(4x) \cos(4x)\,dx\] \[\int_1^e \frac{\ln x - 1}{x}\,dx\]