Integration by Substitution Discussion

(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 one of several techniques for finding antiderivatives that go beyond simply inverting differentiation rules. Section 5.5 in our textbook presents the method in detail. This Canvas discussion is a place for you to practice it, ask questions, and generally start to become comfortable with it.

See which of the following antiderivatives you can find using substitution. (Items 1, 3, and 4 are already done in the class notes from November 23.)

1. \[\int (x-4)^{10}\,dx\]
2. \[\int (3x - 4)^{10}\,dx\]
3. \[\int x \cos \left(x^2\right)\,dx\]
4. \[\int \frac{2x}{x^2+4}\,dx\]

Here are some more challenging antiderivatives that can also be evaluated by substitution. See what you can do with them....

1. \[\int \sec^2(4x)\tan(4x)\,dx\]
2. \[\int \frac{x-1}{x+1}\,dx\]
3. \[\int x e^{2x^2-1}\,dx\]

And here’s one that uses trigonometric identities and substitution: try evaluating