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Chapter 5: Q. 2TB (page 463)

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.
$f (x) = \sec^{- 1} x$

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given,

$f (x) = \sec^{- 1} x$

Step 2.

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Most popular questions from this chapter

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 $x = 4 \sec u$ .

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.

Solve the integral: $\int x \ln (x^{2}) d x$

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.

$2 {(x + 2)}^{2}$

Why doesn’t the definite integral $\int_{2}^{3} \sqrt{1 - x^{2}} d x$ make sense? (Hint: Think about domains.)

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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. f(x)=sec−1⁡x

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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.
$f (x) = \sec^{- 1} x$