Chapter 5: Q.10 (page 477)
Draw pictures to illustrate why the comparison test for improper integrals makes intuitive sense for both convergence comparisons and for divergence comparisons.
1
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Solve the integralthree ways:
(a) with the substitution followed by back substitution;
(b) with integration by parts, choosing localid="1648814744993"
(c) with the trigonometric substitution x = sec u.
For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withor ? Why do we need to think about the unit circle after trigonometric substitution with ?
Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.
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