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The integral \(\Large\int\limits_{1}^{\infty}\frac{dx}{x(x+1)}\)

A) log 2
B) log 3
C) log 5
D) log 7

Correct Answer:

A) log 2

Description for Correct answer:
\(\Large \int\limits_{1}^{\infty}\frac{dx}{x(x+1)}=\lim\limits_{t\rightarrow \infty}\int\limits_{1}^{t}\frac{dx}{x(x+1)}\)

\(\Large =\lim\limits_{t\rightarrow \infty}\int\limits_{1}^{t} \left(\frac{1}{x}-\frac{1}{x+1}\right)dx \)

\(\Large=\lim\limits_{t\rightarrow \infty}\left[ log x-log(1+x) \right]_{1}^{t}\)

\(\Large =\lim\limits_{t\rightarrow \infty}\left[ log\frac{x}{x+1} \right]_{1}^{t}\)

\(\Large =\lim\limits_{t\rightarrow \infty}\left[ log \left(\frac{1}{1+\frac{1}{t}}\right)-log\frac{1}{2}\right]\)

\(\Large =log1-log\frac{1}{2}=-log\frac{1}{2}\)

\(\Large =log2\)

Therefore the integral converges to \(log2.\)

Part of solved Real Analysis questions and answers : >> Elementary Mathematics >> Real Analysis

Comments

Similar Questions

1). Examine the convergence of \(\Large\int\limits_{0}^{\pi}\frac{dx}{1+cosx}\).

A). convergent

B). converges to 1

C). converges to 0

D). diverges

2). Let \(f(x)=x,\ 0\le x\le 1\) and \(p=0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1\) be the partition of [0, 1]. Find U(p, f) and L(p, f).

A). \(\Large \frac{1}{8},\frac{1}{8}\)

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D). \(\Large \frac{7}{8},\frac{9}{8}\)

3). Let f(x) be defined on [0, 1] as follows:\[ f(x) = 1\begin{cases}\text{1 when } x \text{ is rational}\\\text{-1 when } x\text{ is irrational}\end{cases}\]

A). Riemann integrable

B). not Rimann integrable

C). continuous

D). none of these

4). Let \(f(x)=\frac{1}{x}\ 0\le x\le 2\ P=\{ 1,1.2,1.4,1.6,1.8,2 \}\) Find \(w(P,f)\)

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A). \(\Large \frac{a}{3},\frac{a}{3}\)

B). \(\Large \frac{a^{2}}{3},\frac{a^{2}}{3}\)

C). \(\Large \frac{a^{3}}{3},\frac{a^{3}}{3}\)

D). none of these

6). Any countable infinite Set is equivalent to a

A). subset

B). proper subset

C). null set

D). none of these

7). \(Q \times Q\) is

A). uncountable

B). countable

C). finitgset

D). infinite set

A). countable

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