Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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4
votes
0answers
44 views

Integral involving modified Bessel function of the second kind

I would like to calculate the closed-form expression for the following integral: $$ I = \int_{0}^{\infty} x^{M}\exp(-\frac{x}{a})K_{\nu}(b\sqrt{1+x})\mathrm{d}x,$$ where $M$, $a$, and $b$ are all ...
2
votes
1answer
86 views

A mysterious limit related to the integral $\int_{0}^{+\infty}\left(1-\frac{\tanh(ax)}{\tanh x}\right)\,dx$

I have to show that the following limit: $$ \lim_{a\rightarrow0}\Big[\sin(\pi a)\int_0^\infty \left(1-\frac{\tanh ax}{\tanh x}\right)dx\Big]=\pi\ln2$$ holds. This problem relates to my previous ...
1
vote
1answer
35 views

Showing the integral $\int_1^\infty \frac{1}{x(x+p)}\,dx$ is convergent for $p$ greater than $-1$.

Can someone help me why this is true: $$\int_1^\infty \frac{1}{x(x+p)}\,dx = \frac{1}{p}\int_1^\infty\left(\frac{1}{x}-\frac{1}{x+p}\right)dx$$
1
vote
2answers
28 views

Interchange of limit operator and $\ln$ function.

$$\lim_{n\to \infty}\ln\left(\frac{1+a^2n^2}{1+n^2}\right)$$ Can someone help evaluate that for me?
1
vote
1answer
49 views

Integration on manifolds and improper integration

Consider the usual concept of integral on a smooth manifold (the one built using partitions of unity). When applied to the usual smooth structure of $\mathbb{R}^n$, does it coincide with the concept ...
1
vote
2answers
36 views

Improper Integral of $xe^{-x}$.

I was working on this problem but I didn't get the right answer, though I can't find my mistake. Here is the question and my attempt: $\int_a^\infty xe^{-x}dx$ evaluate. $\lim_{b\to \infty} ...
1
vote
0answers
17 views

Clarification of the idea of notations used in integral test proof

I'm looking through some notes presented on the Integral test proof and have been confused over the use of the notations and the concepts associated with the use of the notations like ...
7
votes
1answer
113 views

Proof of an integral identity involving $\pi$ and e

In the "Surprising Identities" post from a while back, Vladimir Reshetnikov offered the following identity[1]: $$\int_{0}^{\infty} dx \frac{1}{1 + x^2} \frac{1}{1 + x^{\pi}} = \int_{0}^{\infty}dx ...
1
vote
0answers
57 views

Holder continuity of the convolution of a Holder continuous function

Let $f(\theta, t)$ be a Holder continuous function for every $t$ on the interval $\theta \in (\alpha,\beta)$. It is known that the application of a singular operator to this function results in ...
2
votes
1answer
31 views

Evaluate : $\lim_{k\to\infty}\int_0^\infty {1\over1+kx^{10}}dx$

Evaluate : $\displaystyle\lim_{k\to\infty}\int_0^\infty {1\over1+kx^{10}} \, dx$. From reading other answer to similar questions, I realized that I may have to use dominated convergence theorem to ...
9
votes
5answers
206 views

Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$

I have found myself faced with evaluating the following integral: $$\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx. $$ Mathematica gives a closed form of $8 \pi ^3/(81 \sqrt{3})$, but I have no ...
3
votes
3answers
71 views

Evaluating the improper integral $\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$

I am supposed to solve this integral but i have no idea how: $$\int_0^1 \frac{\log (x \sqrt{x})}{\sqrt{x}} \,dx$$ Since one limit is $0$ it will be divided by zero. Can someone please explain ...
6
votes
2answers
315 views

How to solve a hard integral?

How prove $ \displaystyle \int _{ 0 }^{ \infty }{ (1+x)\arctan { (x) } } \log^4 { (x) }{\frac{1}{\sqrt{x}(1+x^2)}} dx=\frac{57\pi^6\sqrt{2}}{64} $ I found this integral using numerical values.I ...
1
vote
1answer
45 views

Integration of this using a multi-dimensional hypergeometric function

I want to try and potentially use a Dirichlet - Hypergeometric Function in order to compute the following integral. I would appreciate some help as I'm stuck on how to go about this is a ...
0
votes
1answer
33 views

Determine whether it is convergent or divergent: $\int_{-1}^0 {\frac{e^{1/x}}{x^3}}dx$

So I was evaluating this improper integral, and found the antiderivative to be $e^{1/x}(1-\frac{1}{x})$. How would I evaluate it from $0$ to $-1$? In other words, what would $\frac{1}{0}$ be? ...
1
vote
2answers
92 views

Various evalutions of $\int_0^\infty \sin x \sin \sqrt{x} \,dx$

I'm looking for various ways to evaluate the integral: $$\int_0^\infty \sin x\sin \sqrt{x}\,dx$$ I'm mainly interested in complex analysis. I can think of a wedge -shaped contour of angle $\pi/4$ but ...
1
vote
0answers
28 views

Question on Newman's proof of the Prime Number Theorem

I am reading through Zagier's exposition of Newman's proof of the prime number theorem and I do not understand one of his arguments when proving his so called Analytic Theorem. This theorem states the ...
0
votes
2answers
43 views

Proof of certain Gaussian integral form

I am having trouble understanding where the following integral form comes from: $$\int_{-\infty}^{\infty} e^{-a x^2 }e^{-bx}=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{a}}$$ I see and understand that the value ...
0
votes
2answers
48 views

Convergence of improper integral $\int_0^1 \frac{x^\alpha}{x+x^2}dx$ for $\alpha>0$

I'm having trouble showing the convergence of the integral in the title, for $\alpha >0$: $$\int_0^1 \frac{x^\alpha}{x+x^2}dx $$ I tried using: $$\int_0^1 \frac{x^\alpha}{x+x^2}dx\leq \int_0^1 ...
0
votes
1answer
34 views

Proving the transform of the Q-function

I have the Gaussian Q-function, given by: and I want to prove that it can be also expressed as: Can somebody help explaining how to obtain the second integral from the first?
1
vote
0answers
46 views

Relation between Nuttal Q-function and Gaussian Q-function

I am trying to express the famous Nuttal Q-function, given as: $$\mathcal Q_{m,n}(p,q)=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt$$ where $m$, $n$, $p$, and $q$ are constants and ...
0
votes
1answer
26 views

finding the free energy of a van der waals gas (integration)

I have the following integral, $\int{ \frac{-nrtV}{(v-nb)^{2}} dV}$ could anyone tell me how to do this?
0
votes
2answers
51 views

Convergence of $\int_0^\infty \frac{\sqrt{x}\sin x}{(e^x-1)\log(1+x)}dx$

Could someone please help me determine wether the following integral converges: $$\int_0^\infty \frac{\sqrt{x}\sin x}{(e^x-1)\log(1+x)}dx$$ I have no idea how to start unfortunately... So any hint ...
1
vote
0answers
28 views

Improper and definitive integral of trigonometric functions involving absolute values

Let $x(t)=10\cos(100t+300°)-5\sin(220t - 50°)$ . It is asked to evaluate the following integrals: $$\int_{-\infty}^\infty |x(t)|^2 dt \text{ and } \frac{1}{T} \int_{-T}^T |x(t)|^2 dt$$ Where $ T$ is ...
1
vote
2answers
30 views

A general formula for a specific improper integral

The integral I'm after is here: The question is a little ambiguous whether it wants a general solution for this, but based on thought, I would guess there are many different answers based on the ...
0
votes
5answers
157 views

(another) Challenging improper integral [closed]

$$\int_0^\infty y^{1/2}e^{-y^3}\,dy$$ It is in the section with the gamma function if that helps. Thanks!
2
votes
3answers
61 views

A challenging improper integral

The integral is $$\int_0^1\frac{dx}{\sqrt{-\ln x}}.$$ Not sure if it helps, but it is in the same problem section as $$\int_0^\infty e^{-x^2}dx.$$
0
votes
1answer
60 views

Improper integral show convergence/divergence

How do I show the convergence/divergence of this improper integral? $$\int_1^{\infty}\frac{3-x-x^2\sin x}{3+x+x^3}\,\mathrm dx$$
2
votes
1answer
61 views

How can i solve $\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau$,

I want to find the value of the integral $$\int_0^t \frac{(t-\tau)^{\frac{1}{2}}}{\tau^{\alpha}}d\tau,$$ where $0<\alpha<1$. Using Mathematica I found the solution to be ...
2
votes
3answers
58 views

integral from 1 to infinity of $\frac{5}{(4x+2)^3}$

I have solved the integral: $$\int_1^\infty{\frac{5}{(4x+2)^3}}dx$$ using u substitution and I am not getting the correct answer. I am missing some step here or making an algebra error. I am not ...
2
votes
1answer
31 views

Functions such that $\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$ and …

Can we find a bounded function $a:\mathbb{R}\to\mathbb{R}$ and a function $f\in L^1(\mathbb{R})$ with $f\neq 0$ such that $$\sup_{t\in\mathbb{R}}\int_{\mathbb{R}}e^{a(x)t}|f(x)|dx<\infty$$ and ...
1
vote
1answer
65 views

$\mu$ test for convergence of improper integral of first kind

While going through an Indian text on Analysis I found a test for convergence of improper integral.It was stated without proof.I tried to prove it..then some doubts pop up... Statement is this :Let ...
9
votes
1answer
230 views

Other integral related to Ahmed's integral

I have a doubt regarding the evaluation of the following integral : $$ \int_0^\frac{1}{\sqrt{5}} \frac{\tan^{-1}\left({\sqrt{(1 + x^2)/2}}\right)} {(1 + 3x^2)\sqrt{1 + x^2}}\,du = ...
0
votes
1answer
36 views

Find the area described by $S=\{(x,y)\mid -2<x\leq 0, 0\leq y\leq 2/\sqrt{x+2}\}$

Find the area described by $S=\{(x,y)\mid -2<x\leq 0, 0\leq y\leq 2/\sqrt{x+2}\}$. I have tried to use logic to figure out what to do but cannot figure it out. I have tried to think about this ...
0
votes
0answers
54 views

question regarding evaluating improper integrals with complex analysis-$\int_0^\infty\sin(x^2)\,dx$ [duplicate]

I'm asked to solve the following integral with complex analysis methods: $$\int_0^\infty\sin(x^2)\,dx$$ The thing that bothers me, is that this integral has no limit in the real plane, because the ...
1
vote
2answers
58 views

what is the integral on $[0,2]$ of $x/(3-2x)$

i know that this is an improper integral, but when you evaluate the limits as $x\to (3/2)^-$ and $x\to (3/2)^+$, you get positive and negative infinity but I am not sure if you can cancel them ...
0
votes
2answers
47 views

Help with an improper integral!

I need some help with an indefinite integral problem (only the $2^{\textrm{nd}}$ part thou). Problem is as follows. Consider the function $f(x) = \dfrac{\ln\!\left(x\right)}{x^{p}}$, where $p>1$ ...
3
votes
3answers
165 views

Integration of $\int_{-\infty}^{\infty} e^{-x^2 + 2x} dx$

The question is easy to phrase: Show that $$ \int_{-\infty}^{\infty} e^{-x^2 + 2x} dx $$ converges and compute its value. A first thing I did is simplifying it to $\int_{-\infty}^{\infty} e^{-y^2 + ...
1
vote
0answers
11 views

Doing integration by parts on exponential families

I have a problem of estimating moments of an exponential family by integration by parts. Lets consider the exponential family in its canonical form. $f(x)=e^{\theta x-\psi(\theta)}h(x)$. The ...
0
votes
1answer
66 views

quirk involving trig substitution?

I have reduced the following trig identity to the following which is correct. $$\int \cos^2(x)\tan^3(x)dx = \int \tan(x) - \sin(x)\cos(x)dx$$ However this next step changes the value of my ...
14
votes
4answers
468 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
5
votes
3answers
80 views

Is $\int_0^\infty x^{a-1} (1-x)^{b-1} e^{t-cx} dx$ integrable?

I am trying to evaluate the integral below. Is it even integrable? (Online integral solvers e.g. WolframAlpha could not solve the indefinite or the definite integral.) $$\int_0^\infty x^{a-1} ...
1
vote
2answers
44 views

Find the volume of this improper integral?

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by $f(x)=\sqrt{\frac{(x+1)}{x^3}}$ and the $x$-axis on the interval $[1,\infty)$ is resolved ...
0
votes
2answers
51 views

Calculate improper integral using Euler's integral

I have to evaluate the following integral $$\int_0^2 \frac{dx}{\sqrt[5]{x^3(2-x)^2}}$$ Thanks in advance.
1
vote
1answer
46 views

Prove equality with product of improper integrals

I have to prove the following equality: $$ \int\limits_0^{+\infty} \frac{dx}{\sqrt{\cosh x}} \cdot \int\limits_0^{+\infty} \frac{dx}{\sqrt{\cosh^3 x}} = \pi. $$ The first integral Wolfram Mathematica ...
0
votes
0answers
54 views

Leibniz Rule in Improper Integrals?

http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf In the above link, you will find a proof and various examples of the Leibniz rule. The rule given applies to integrals with finite ...
0
votes
0answers
21 views

About the special integral forms of bessel functions

Prove that according to http://people.math.sfu.ca/~cbm/aands/page_360.htm and http://people.math.sfu.ca/~cbm/aands/page_376.htm, $Y_0(z)=\dfrac{4}{\pi^2}\int_0^\frac{\pi}{2}\cos(z\cos ...
0
votes
0answers
29 views

About the extensions from Confluent Hypergeometric Function of the Second Kind

I know that $\int_0^\infty t^{a-1}(1+t)^{c-a-1}e^{-yt}~dt=\Gamma(a)U(a,c,y)$ , where $\text{Re}(a),\text{Re}(y)>0$ . How about $\int_0^\infty t^{a-1}(1+t)^{c-a-1}(1+xt)^{-b}e^{-yt}~dt$ and ...
0
votes
1answer
54 views

how do i know when to reevalute limits on indefinite integral?

A) $$\int_0^{ln(3)} \frac{e^x}{e^x +2}dx$$ $$u=e^x +2$$ $$du = e^x$$ $$\int_0^{ln(x)} u^{-1}du$$ when x = ln(3), u = 5 ...
-2
votes
3answers
79 views

improper integral of exponential function [duplicate]

I have a problem calculating improper integrals, this one for example, can you please help me solve it? $$\int_0^\infty t^3(e^{-t^2})dt$$ thanks in advance.