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Chapter 5: Q. 36 (page 417)
Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
The solution of the given integral is .
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Find three integrals in Exercises 39鈥�74 that can be solved without using trigonometric substitution.
Show that if , then , in the following two ways: (a) by using implicit differentiation, thinking of as a function of , and (b) by thinking of as a function of .
Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) An integral with which we could reasonably apply trigonometric substitution with .
(b) An integral with which we could reasonably apply trigonometric substitution with .
(c) An integral with which we could reasonably apply trigonometric substitution with .
Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.
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Give an example of an integral for which trigonometric substitution is possible but an easier method is available. Then give an example of an integral that we still don鈥檛 know how to solve given the techniques we know at this point.
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