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Chapter 5: Q. 40 (page 478)

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

Short Answer

Expert verified

The value of the integral is ${鈭�}_{0}^{鈭�} \frac{x}{{(x^{2} + 1)}^{2}} d x = \frac{1}{2} .$

Step by step solution

01

Step 1. Given information.

The given integral is the following.

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

02

Step 2. Value of integral.

Solve the integral by using the definition of improper integral.

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

The value of the integral is ${鈭�}_{0}^{鈭�} \frac{x}{{(x^{2} + 1)}^{2}} d x = \frac{1}{2} .$

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Most popular questions from this chapter

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 the integral: $鈭� \frac{x^{2}}{e^{x}} d x$

Solve the integral: $鈭� \ln (3 x) d x$

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.

$鈭� \frac{x^{2} + 9}{x^{\frac{3}{2}}} d x$

Solve the integral

$鈭� \frac{1}{\sqrt{x^{2} + 9}} d x$

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