Chapter 5: Q. 38 (page 417)
Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)
Short Answer
The solution of the given integral is .
Chapter 5: Q. 38 (page 417)
Solve each of the integrals in Exercises 21–70. 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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Get started for freeWhy doesn’t the definite integral make sense? (Hint: Think about domains.)
Domains and ranges of inverse trigonometric functions: For each function that follows, (a) list the domain and range, (b) sketch a labeled graph, and (c) discuss the domains and ranges in the context of the unit circle.
Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.
True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: The substitution x = 2 sec u is a suitable choice for solving.
(b) True or False: The substitution x = 2 sec u is a suitable choice for solving.
(c) True or False: The substitution x = 2 tan u is a suitable choice for solving
(d) True or False: The substitution x = 2 sin u is a suitable choice for solving
(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form .
(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.
(g) True or False: When using trigonometric substitution with , we must consider the cases and separately.
(h) True or False: When using trigonometric substitution with , we must consider the cases and separately.
For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
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