Chapter 5: Q. 6 (page 417)
For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
Short Answer
The three integrals will have form after a substitution of variables.
Chapter 5: Q. 6 (page 417)
For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
The three integrals will have form after a substitution of variables.
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Get started for freeFor each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.
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 .
Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Then
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.
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.
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