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Chapter 5: Q. 26 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises.
${鈭�}_{0}^{1} x^{- 1.01} d x$

Short Answer

Expert verified

The improper integral on $[0, 1]$ diverges.

${鈭�}_{0}^{1} x^{- 1.01} d x = 鈭�$

Step by step solution

01

Step 1. Given information.

The given improper integral is following.

${鈭�}_{0}^{1} x^{- 1.01} d x$

02

Step 2. Value of integral.

Write the integral in the proper form of ${鈭�}_{0}^{1} \frac{1}{x^{p}} d x .$

${鈭�}_{0}^{1} \frac{1}{x^{p}} d x = {鈭�}_{0}^{1} x^{- 1.01} d x {鈭�}_{0}^{1} \frac{1}{x^{p}} d x = {鈭�}_{0}^{1} \frac{1}{x^{1.01}} d x p = 1.01 & 1.01 > 1$

The Fundamental Theorem of Calculus tells us that the improper integral on $[0, 1]$ diverges.

${鈭�}_{0}^{1} x^{- 1.01} d x = \lim_{a 鈫� 0} {鈭�}_{a}^{1} x^{- 1.01} d x = \lim_{a 鈫� 0^{+}} {[- 100 x^{- 0.01}]}_{a}^{1} = \lim_{a 鈫� 0^{+}} (- 100 + 100 a^{- 0.01}) = 鈭�$

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Most popular questions from this chapter

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Solve given definite integral.

${鈭�}_{1 / 4}^{1 / 2} 鈥� \frac{1}{x^{2} \sqrt{1 鈭� x^{2}}} d x$

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� \frac{1}{{(x^{2} + 4)}^{\frac{3}{2}}} d x$

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$2 x^{2} 鈭� 4 x + 1$

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 $鈭� \frac{1}{x^{2} 鈭� 4} d x$ .

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

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving $鈭� \frac{1}{x^{2} + 4} d x .$

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

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

(f) True or False: Trigonometric substitution doesn鈥檛 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.

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