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Chapter 5: Q. 42 (page 407)

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21鈥�56.
${鈭�}_{0}^{鈭�} \frac{x}{(x^{2} + 1)} d x$

Short Answer

Expert verified

The value is $鈭�$ .

Step by step solution

01

Step 1. Given information.

The given function is ${鈭�}_{0}^{鈭�} \frac{x}{(x^{2} + 1)} d x$ .

02

Step 2. Value of the integral.

The given integral can be written as,

${鈭�}_{0}^{鈭�} \frac{x}{(x^{2} + 1)} d x = \lim_{B 鈫� 鈭�} {鈭�}_{0}^{B} \frac{x}{(x^{2} + 1)} d x$

$Let u = x^{2} + 1 i.e d u = 2 x d x T h e r e f o r e, 鈭� \frac{x}{(x^{2} + 1)} d x = 鈭� \frac{1}{2 u} d u = \frac{1}{2} 鈭� \frac{1}{u} d u = \frac{1}{2} \ln | u | 鈭� \frac{x}{(x^{2} + 1)} d x = \frac{1}{2} \ln |x^{2} + 1|$

03

Step 3. Substitution.

Substituting the obtained value in the given equation, we get,

$鈭� \frac{x}{(x^{2} + 1)} d x = \frac{1}{2} \ln |x^{2} + 1| {鈭�}_{0}^{鈭�} \frac{x}{(x^{2} + 1)} d x = \lim_{B 鈫� 鈭�} \frac{1}{2} \ln |x^{2} + 1| = 鈭� - 0 = 鈭�$

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Most popular questions from this chapter

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.

$鈭� \frac{\sqrt{4 - x^{2}}}{x^{2}} d x$

Solve the integral $鈭� x^{3} \sqrt{x^{2} - 1} d x$ three ways:

(a) with the substitution $u = x^{2} - 1,$ followed by back substitution;

(b) with integration by parts, choosing localid="1648814744993" $u = x^{2} and d v = x \sqrt{x^{2} - 1} d x;$

(c) with the trigonometric substitution x = sec u.

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.

$x^{2} - 5 x + 1$

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

$(a) 鈭� \frac{4 + x^{2}}{x} d x$ $(b) 鈭� \frac{x}{4 + x^{2}} d x$

role="math" localid="1648759296940" $(c) 鈭� \frac{x^{2}}{4 + x^{2}} d x$ $(d) 鈭� \frac{16 鈭� x^{4}}{4 + x^{2}} d x$

Solve given definite integral.

${鈭�}_{0}^{4} 鈥� x^{3} \sqrt{x^{2} + 4} d x$

See all solutions

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