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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5
votes
3answers
685 views

Improper integral of a rational function!

Find the value of the integral $$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx$$ I tried the substitution $x=t^5$ to obtain $$\int_0^\infty \frac{5t^6}{1+t^{10}}dt$$ Now we can factor the denominator to ...
1
vote
1answer
51 views

Value of $\int_{-\infty}^{\infty} \delta(t-\pi)\cos(t) \,dt$

What is the value of $$\int_{-\infty}^{\infty} \delta(t-\pi)\cos(t) \,dt?$$ I calculated the value to be infinity but I need to use the definition of the dirac delta function to prove this but I am ...
1
vote
2answers
105 views

Evaluate $\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$

This is a integration question from a previous calculus exam: Evaluate $$\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$$ I rewrote it as $$\lim \limits_{b \to \infty} \int_0^b ...
5
votes
2answers
164 views

$\int_{0}^{\infty}e^{-st}h(t)dt=0 \Rightarrow h(t)=0.$

Suppose $h(t)$ is continuous function and $\int_{0}^{\infty}e^{-st}h(t)dt=0 ~\forall~ s>s_{0}$, then prove that $h(t)=0$. I know "if a function is continuous, non-negative or non-positive, and its ...
1
vote
1answer
59 views

Prove $\int {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= {\operatorname{sinc}}(\lambda-\nu ).$

I want to prove the following relation. For any real numbers $\lambda$ and $\nu$, we have \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) ...
3
votes
5answers
123 views

How to show $\int_0^1 \left( \frac{1}{\ln(1+x)} - \frac{1}{x} \right) \,dx$ converges?

I need to show that $$\int_0^1 \left( \frac{1}{\ln(1+x)} - \frac{1}{x} \right) \,dx$$ converges, given that $$\lim_{x\rightarrow0^+} \left( \frac{1}{\ln(1+x)} - \frac{1}{x} \right) = \frac{1}{2}$$ ...
2
votes
4answers
338 views

Where am I wrong in the following limit?

We have this function: $f(x)=\frac{2x+3}{x+2}$ and we need to find this: $$\lim _{x\to \infty \:}\frac{\int _x^{2x}f(t)\,dt}{x}$$ Now I will tell how I solved this: I suppose that $$\int _x^{2x} f(t) ...
4
votes
1answer
57 views

Counter-example to $\int_0^\infty f(x) dx=\lim_{t\to\infty} \int_{1/t}^t f(x) dx$

I want to prove or disprove the statement that, for a function $f$ that is continuous on $(0,\infty)$, we have $\displaystyle{\int_0^\infty f(x)\ dx=\lim_{t\to\infty} \int_{1/t}^t f(x)\ dx}$. My ...
5
votes
5answers
391 views

Proving that $\int_{-\pi}^{\pi} \ln |1 - e^{i\theta}| d\theta = 0$

I found this on some comprehensive exam. Prove that $\int_{-\pi}^{\pi} \ln |1 - e^{i\theta}| d\theta = 0$. I was wondering would standard approach work? By that I just mean splitting the ...
2
votes
1answer
122 views

Asymptotic behavior of the confluent hypergeometric function

Consider the following function $$U(a,z)= \frac{1}{\Gamma(a)} \int^{\infty}_0 t^{a-1} \cdot (1+t)^{-a} e^{-zt} dt$$ My Try : Let $\tau= zt$, then : $$ U(a,z)= \frac{z^{-a}}{\Gamma(a)} \int^{\infty}_0 ...
1
vote
0answers
60 views

Hints to find analytical solution to integral

I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - ...
0
votes
1answer
38 views

An explanation of the integration

So, the integral is: $$\int_1^2\frac{x-2}{\sqrt{x-1}}dx$$ If I copied correctly from the board, the teacher said if x approaches 1+, the function approaches +$\infty$. What is the difference between ...
2
votes
1answer
65 views

How to solve this improper integral? [duplicate]

The problem is: If $f(x)\in C[0,+\infty)$, $\displaystyle\lim_{x\to+\infty}f(x)=k\in\mathbb R$, and $b>a>0$, prove: $$\int_{0}^{+\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-k]\ln(\frac ba)$$ My ...
0
votes
1answer
101 views

Is this Brownian Integral identity correct?

$$\int_0^1 B_t dt=\lim_{\omega \to\infty}{1 \over {\omega}}{\int_0^{\omega}{Y_0+}X_t dt}$$ Where $B_t$ is simple brownian motion, and $X_t$ is a discrete random variable that can be 1 or -1 with ...
0
votes
1answer
45 views

Is there another way than linearization?

$$I= \int {\sin^mx \cos^nx }dx$$ I need a Hint on doing this integral a Successive Partial Integration but it seems that the problem shows up when $ m = 2k $ and $ n = 2p$ where $p,m \in \mathbb{N}$. ...
2
votes
4answers
84 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
1
vote
2answers
87 views

Closed form of the integral $\int_0^1 \frac{x^n}{1+x}\, dx$

I am trying to evaluate the integral $$\int_0^1 \frac{x^n}{1+x}\, dx, \;\;\; n \in \mathbb{N}$$ in a closed form. I tried tackling it using Beta Form $\displaystyle \int_0^1 ...
0
votes
2answers
52 views

Convergence of the improper integral $\int_{0+}^{1-} \frac{\log x}{1-x} dx$

Let $0 < t_{1} \leq t_{2} < 1.$ Then $$\int_{t_{1}}^{t_{2}} \frac{\log x}{1-x} dx = \int_{1/t_{2}}^{1/t_{1}} \frac{\log u^{-1}}{1 - u^{-1}}(-u^{-2}) du = \int_{1/t_{2}}^{1/t_{1}} \frac{\log ...
4
votes
1answer
101 views

Prove that $ f:(a,b)\to\mathbb{R}$ is integrable iff $\lim_{\epsilon\to0} \int_{[a+\epsilon,b-\epsilon]}f$ exists

I want to solve the following: Let $ f:(a,b)\to\mathbb{R}$ continous such that $f(x)\ge 0 $ for all $x\in(a,b)$. Show that $f$ is integrable iff $\displaystyle \lim_{\epsilon\to0} ...
5
votes
3answers
183 views

Improper Integral of a periodic function converges

Given $f(x)$ is a periodic function and $\int_0^p{f(x)}dx=0$. Show $\int_1^\infty\frac{f(x)}{x}dx$ converges. 1) I know this integral can be broken into ...
0
votes
0answers
36 views

improper integral, convergent?

Fix a parameter $\theta \in (0,1/2]$. I am trying to figure out for which values of the parameter $\alpha\in \mathbb{R}$ this integral converges. $$\int_0^1 \frac{(1-(1-x)^{-\theta})^2}{x^\alpha} ...
2
votes
1answer
72 views

Using Laplace Transforms to Evaluate Integrals

I'm trying to solve $$\int_0^{\infty} \, \frac{e^{-2t}\cos(3t)-e^{-4t}\cos(2t)}{t}dt.$$ I'm sure that this involves Laplace transforms, I'm just not sure how. I would start by separating and ...
1
vote
0answers
51 views

uniform convergence and improper integral(convolution)

I want to show that $$\frac{d}{dx}(f*g)=(\frac{d}{dx}f)*g$$ where $f(x)=\frac{1}{\sqrt{x}}e^{-\frac{1}{x}}$, $g(y)$ is continuous and bounded. the convolutions are improper integrals. I'm now here ...
11
votes
1answer
210 views

How to find the value of $I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$

How to find the value of $$I_1=\int_0^\infty\frac{\sqrt{x}\arctan{x}\log^2({1+x^2})}{1+x^2}dx$$ If we put $$I_2=\int_0^\infty\frac{\arctan^2({x})\log({1+x^2})}{\sqrt{x}(1+x^2)}dx$$ After long ...
0
votes
1answer
33 views

Substitution in integral, how shall I proceed

Say we have $\int_2^\infty \frac{1}{(\log n)^{\log n}}dn.$ Let $u=\log n.$ We have the boundaries become $u=\log 2$ and $u=\infty.$ How should I proceed with $dn.$ I have $du=\frac{1}{n}dn,$ hence, ...
0
votes
1answer
61 views

Transform an Integral bounds from -inf, inf to 0 to 1

Good day, If i have an integral from -infinity to infinity, how do I change the bounds/limits to 0 to 1? I don't want to give exact question since this is part of an assignment. I know how to figure ...
4
votes
0answers
83 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 ...
3
votes
1answer
102 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
36 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
51 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
103 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
42 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
25 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
161 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
110 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
32 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
230 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
77 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 ...
9
votes
2answers
764 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
49 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
56 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
107 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
40 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
74 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
57 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
90 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
68 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
65 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
60 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
33 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 ...