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Chapter 5: Q. 2 (page 463)

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 $x = \tan u$ .
(b) An integral with which we could reasonably apply trigonometric substitution with $x = 4 \sec u$ .
(c) An integral with which we could reasonably apply trigonometric substitution with $x - 2 = 3 \sin u$ .

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Most popular questions from this chapter

List some things which would suggest that a certain substitution u(x) could be a useful choice. What do you look for when choosing u(x)?

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 $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(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 $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

Find three integrals in Exercises 27–70 for which a good strategy is to use integration by parts with $u = x$ and dv the remaining part.

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial $x^{2} - x + 3$ as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Then $p (x) =___a n d \frac{p (x)}{q (x)} =____$

What is a rational function? What does it mean for a rational function to be proper? Improper?

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