0
votes
0answers
19 views

Generalized and Lebesgue Integrable Function

I am reading a chapter about "Generalized Riemann Integrals" from Introduction to Real Analysis by Bartle & Sherbert. I have just finished reading section 10.2 which is about "Improper and ...
4
votes
1answer
65 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
0
votes
1answer
62 views

How to prove that integral of function is convergent

$\int_{0}^{\infty} \frac{(\sin(x) )}{ x} \,\mathrm dx$ and $\int_{0}^{\infty} \frac{(\sin(x) \arctan(x))}{ x} \,\mathrm dx$ These are convergent. How to prove that?? I using the comparison test. ...
2
votes
2answers
60 views

a question about summation of series, how to prove $\int_0^\infty e^{-x}S(x)$=$\sum_{i=0}^\infty a_nn!$

If the coefficients of $\sum_{n=0}^\infty a_nx^n$ is non-negative($a_n\ge 0$ for every n),and the sum function is S(x). Also,suppose$\sum_{i=0}^\infty a_nn!$ is convergent,please prove $\int_0^\infty ...
5
votes
5answers
82 views

a question about a complex integral, I am struggling with it!

How to prove $$\int _0^1 {\ln(x)\over{1-x^2}}={-\pi^{2}\over 8}$$ My solution: If we can prove$\int _0^1 {\ln(x)\over{1-x^2}}= \lim_{n\to \infty} \int _0^1\ln(x)(1+x^2+x^4+......+x^{2n})$,then I ...
17
votes
1answer
325 views

Integral $\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}$

Please help me prove the following identity: $$\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}\quad a>0$$ This integral is from the ...
5
votes
1answer
89 views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$

I'm trying hard to solve this integral but I still don't know how... $$\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$$ The integral from $-\infty$ to $\infty$ is quite easy, but how could we integrate ...
0
votes
1answer
60 views

a question about integral proof: $\lim_{n\to \infty} \int_{0}^\infty {n\cdot {\ln(1+{f(x)\over n}}})dx=\int_{0}^\infty f(x)dx$

A non-negative function ${\rm f}\left(x\right)$ is continuous in $(0,\infty)$ and $\displaystyle{\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x}$ is convergent. Then, we need to prove $$\lim_{n\to ...
3
votes
2answers
62 views

How to prove divergence of the integral $\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$

I want to show that the following integral diverges: $$\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$$ I used the substitution $ t = \frac{1}{x} $ to transform this integral into $$\int_{0}^{\infty}\sin ...
1
vote
1answer
66 views

Show that the improper integral $\int_1^\infty f(x) \ dx$ exists iff $\sum_1^\infty a_n$ converges.

The assignment: Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and $f: [1, \infty) \rightarrow \mathbb{R}$ be a function, defined by $f(x) = a_n$, for $x \in [n,n+1).$ Show that: ...
1
vote
0answers
21 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
0
votes
0answers
29 views

Need help on analytically proving the monotonicity of an inexplicit integral

I have the following function which I have numerically investigated to be monotonically increasing in $\nu$: $D(x;\mu,\nu) =\frac{1}{\sqrt{2\pi \cdot \nu}}\int_{-\infty}^{\infty}e^{-\theta \cdot x} ...
1
vote
2answers
78 views

a question about improper integral, I cannot solve it!

If $f(x)$ is continuous in $[0,\infty)$, and $\displaystyle\int_c^{\infty}\frac{f(x)}{x}\, dx$ is convergent for any $c>0$. Please prove $\displaystyle\int_0^{\infty}\frac{f(\alpha x)-f(\beta ...
1
vote
2answers
54 views

Can somebody help me solve the proof about improper integrals?

If $f(x) > 0$ is continuous at $[0, +\infty]$ and $\displaystyle \int_0^{+\infty} \frac{1}{f(x)} dx$ is convergent, please prove $\displaystyle \lim_{\lambda \to \infty} \frac{1}{\lambda} ...
1
vote
2answers
49 views

Can an integral be proved to have a finite value if an upper bound of the integrand has a finite value for improper integrals?

Can we say $ \int_{0}^{\infty} f(x) \text{dx} < \infty$ if $\exists \quad g(x) : \quad g(x)\geq f(x)\; \forall x \in \mathbb{R}$ and $ \int_{0}^{\infty} g(x) \text{dx}$ is finite. If yes, ...
0
votes
4answers
127 views

Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }} $$ ...
1
vote
1answer
48 views

Show $\int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0$

let $\rho(x)=\sqrt{x}, \hspace{4mm} \forall x \in \mathbb{R}$ Show : $$ \int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0. $$ My attempt: \begin{align*} ...
1
vote
3answers
76 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
3
votes
2answers
109 views

Improper parametric integral and differentiation under the integral sign

While looking at an astrophysic problem, I encountered the following integral $$ \rho_{\infty} (r) = \int_{r}^{a} \frac{\rho_{0} (r_{0})}{\sqrt{r_{0}^{2} - r^{2}}} d r_{0} \;\;\;\;\;\;\; (1)$$ The ...
1
vote
0answers
55 views

Improper Riemann integral and imaginary exponential of real polynomials

Let $P(x_1,\cdots,x_n)$ be a real polynomial of degree $\geq 2$. What are the conditions on $P$ so that $$ I_P:=\int_{\mathbb{R}^n} e^{iP(x)} dx $$ exists as an improper Riemann integral ? Already ...
0
votes
1answer
48 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
2
votes
1answer
110 views

Evaluation of a definite integral

I want to find the best way to show $\int_0^\infty\dfrac{x^{2m}}{x^{2n}+1}\,dx=\dfrac{\pi}{2n}\operatorname{csc}\left(\dfrac{2m+1}{2n}\pi\right)$, where $0\leq m<n$. It's easy to verify some ...
0
votes
1answer
75 views

How to find this limit $\lim_{x\to+\infty}f(x)=0$

let $f(x)\in C^{1}[0,+\infty)$ , and such improper integral $$\int_{0}^{+\infty}\left(|f(x)|+|f'(x)|\right)dx$$ is convergence show that $$\lim_{x\to+\infty}f(x)=0$$ My try: since ...
2
votes
3answers
89 views

$\int_0^\infty \frac{x^p}{1+x^p}dx$

Let $p\geq -1$. How do I show that $\int_0^\infty \frac{x^p}{1+x^p}dx$ diverges? I thought to break up the integral into $\int_0^1 \frac{x^p}{1+x^p}dx+\int_1^\infty \frac{x^p}{1+x^p}dx$, but I ...
2
votes
2answers
92 views

Why does $\int_0^ae^\frac1x x^pdx$ diverge?

Let $a$ be positive and $p$ be real. Why does the improper integral $$\int_{0}^{a}{\rm e}^{1/x} x^{p}\,{\rm d}x$$ diverge ? Direct integration over $\left[b,a\right]$ for positive $b$ is hard. On ...
1
vote
4answers
247 views

$\int_1^\infty\frac 1{\ln x}dx$

How do I show that $\int_1^\infty\frac 1{\ln x}dx$ diverges? I'm thinking to break up the integral into two parts, say, over $[1,2]$ and $[2,\infty)$, but how do I integrate the integrand? I'm ...
0
votes
2answers
95 views

Some standard improper integrals

Referring to part (b): I don't know what $a$ stands for, and also don't get what "convergence" the author is referring to here. If he means the convergence as $a$ tends to $\infty$, then shouldn't ...
1
vote
1answer
289 views

Why does Fubini's theorem not apply in this example?

Let $f(x,y)=\frac{x-y}{(x+y)^3}$ and $g(x,y)=\text{sgn}(x-y)e^{-|x-y|}$ ( http://en.wikipedia.org/wiki/Sign_function ) We have $$\int_0^1\int_0^1f(x,y)dy dx = 1/2 = -\int_0^1\int_0^1f(x,y)dy dx$$ ...
1
vote
1answer
135 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
3
votes
3answers
168 views

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.
4
votes
3answers
160 views

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$ The answer given by Wolfram Alpha is $\sqrt{2\pi}\xi\exp(-\xi^2/2)$. Observe how this is related to the Fourier transform of ...
3
votes
4answers
172 views

Integral from zero to infinity of $\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda$

I know that the value of the integral is as follows $$\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda =z^a \frac{\Gamma(1-a)}{a}$$ However, how exactly the integral is calculated? ...
1
vote
1answer
42 views

Calculation of an integral via residue.

$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$ How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked ...
1
vote
2answers
123 views

$\dfrac{\sin x}{x}$ modified improper integrals.

I am trying to evaluate this integrals: $$ \int_{-\infty}^{\infty} \! \left[\frac{\sin\left(x\right)}{x}\right]^n \, \mathrm{d}x. $$ I know how to prove it if $n=1$ using Fourier Transform, but I ...
1
vote
0answers
62 views

How to prove that this integral converges absolutely?

$f:[a,{\infty}[\to\mathbb{R}\ $ is bounded and suppose that $f$ is integrable on each interval of the form $[a,b[$. Prove that $$\int_0^\infty \frac { f(x) }{ x^p } \ \, dx$$ converges absolutely ...
14
votes
1answer
271 views

Proving a formula for $\int_0^\infty \frac{\log(1+x^{4n})}{1+x^2}dx $ if $n=1,2,3,\cdots$

I came across the formula $$\int_0^\infty \frac{\log \left(1+x^{4n} \right)}{1+x^2}dx = \pi \log \left\{2^n \prod_{k=1 ,\ k \text{ odd}}^{2n-1} \left(1+\sin \left( \frac{\pi k}{4n}\right) ...
12
votes
2answers
593 views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$ We can prove using the Beta-Function identity that $$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...
1
vote
3answers
497 views

How to determine whether an integral is convergent

I missed up the last lecture and can't understand how to determine whether an integral with parameters is convergent or divergent? For example: For which values of the parameters $p,q \in ...
4
votes
1answer
133 views

Analysis on Improper Integrals

This question is from Munkres' Analysis on Manifolds, section 15 question 1. Let $f: \mathbb{R} \to \mathbb{R}$ be the function $f(x) = x$. Show that, given $\lambda \in \mathbb{R}$, there exists a ...
1
vote
2answers
316 views

Uniform convergence of integral on unbounded interval

Suppose I have $\;f_n:[a,\infty)\to\mathbb{R}$ and $\int_a^\infty f_n(x) dx$ exists. If $f_n\to f$ uniformly on $[a,\infty)$, I am able to show that $\int_a^\infty f(x)dx$ exists and $\int_a^\infty ...
2
votes
2answers
29 views

Problems with determining convergence of integral

It should be easy but I'm not sure... For which $\alpha \in \mathbb{R}$ the following integral is convergent: $$\int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha}dxdy \ \ ?$$ I get for all $\alpha \neq 1,2$ ...
3
votes
1answer
205 views

Integration of hyperreal functions / Intermediate Value Theorem

Here's a statement on hyperreal function I've been trying to prove (I came up with it but I think it is true): Suppose $f(x)$ is a continuous real-valued function and $h(x)$ is a continuous ...
2
votes
1answer
172 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
3
votes
2answers
148 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
4
votes
3answers
641 views

Why isn't an odd improper integral equal to zero

My calculus book says that the integral of $\frac1x$ cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and ...
2
votes
1answer
144 views

Assess the limit: $ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$

Compute the following limit: $$ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$$ I'm looking for an easy approach if possible.
4
votes
1answer
3k views

Can we integrate discontinuous functions

Does it make sense to integrate the function $f(x) = 6$ for $x\in [0,1)$ and $f(x) = 1/x$ for $x$ in $(1,\infty)$? Does it make sense to integrate the function $f(x) = 6$ for $x\in [0,1]$ and $f(x) = ...
15
votes
3answers
353 views

Computing $ \int_{0}^{\infty} \frac{1}{(x+1)(x+2)…(x+n)} \mathrm dx $

I would like to compute: $$ \int_{0}^{\infty} \frac{1}{(x+1)(x+2)...(x+n)} \mathrm dx $$ $$ n\geq 2$$ So my question is how can I find the partial fraction expansion of $$ ...
2
votes
3answers
340 views

Improper Integrals

I don't get how we're supposed to use analysis to calculate things like: a) $$ \int_0^1 \log x \mathrm dx $$ b) $$\int_2^\infty \frac{\log x}{x} \mathrm dx $$ c) $$\int_0^\infty \frac{1}{1+x^2} ...
63
votes
18answers
13k views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin x}{x} \ dx = \frac{\pi}{2}$$ Well, can anyone ...