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Chapter 5: Q. 44 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21鈥�56.
${鈭�}_{0}^{蟺 / 2} \tan x d x$

Short Answer

Expert verified

The value is $鈭� (D i v e r g e n t) .$

Step by step solution

01

Step 1. Given Information.

The given integral is ${鈭�}_{0}^{蟺 / 2} \tan x d x$ .

02

Step 2. Value of the integral.

On solving,

$= \lim_{b 鈫� {\frac{蟺}{2}}^{-}} ({鈭�}_{0}^{b} \tan x d x) = \lim_{b 鈫� {\frac{蟺}{2}}^{-}} (\ln | \sec x |]_{0}^{b} = \lim_{b 鈫� {\frac{蟺}{2}}^{-}} (\ln | \sec b | - \ln | \sec 0 |) = \ln (鈭�) - 0 = 鈭� = D i v e r g e n t .$

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Most popular questions from this chapter

Solve given definite integrals.

${鈭�}_{0}^{4} x \sqrt{x^{2} + 4} d x$

Why doesn鈥檛 the definite integral ${鈭�}_{2}^{3} \sqrt{1 - x^{2}} d x$ make sense? (Hint: Think about domains.)

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

$鈭� \frac{x^{3} - 3 x^{2} + 2 x - 3}{x^{2} + 1}$ dx

Explain how to know when to use the trigonometric substitutions $x = a \sin 鈦� u, x = a \tan 鈦� u, a n d x = a \sec 鈦� u$ , Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for $x$ and $u$ in each case?

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.

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