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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11
votes
6answers
743 views

Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
1
vote
1answer
40 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
0
votes
1answer
349 views

Improper integral of a function involving square root and absolute value.

$$\int_{-2}^{8}\dfrac{dx}{\sqrt{|2x\|}}$$ I understand that you have to split this into two integrals because at $x=0$, the function is not defined. The example showed that they split up the integral ...
3
votes
0answers
40 views

Evaluation of $\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d\!}x$ for $\mu>0$

I'm trying to evaluate the integral $$ \Psi(\mu,\nu)=\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d}\!x\qquad(\text{for}\; \mu>0)\tag 1 $$ where $\nu\in\Bbb R$ and $f$ ...
1
vote
1answer
60 views

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) - \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} ...
0
votes
3answers
64 views

General closed form of $\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos \theta_o}} d\theta$

I once asked a question about how to integrate the reciprocal of the square root of cosine. Is there a general closed form for the integral $$\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos ...
0
votes
1answer
54 views

How to check if functions are integrable?

Consider two functions $$ \int_0^1 \frac{1}{e^x-1} dx $$ and $$ \int_0^1 \frac{1}{(e^x-1)^2} dx $$ How to check if these functions are integrable?
0
votes
1answer
22 views

Discussing the convergence of $\displaystyle\int_I\frac{x+2}{\sqrt x\left(x^2+x+1\right)^4}\mathrm dx$

Let $$f(x) = \frac{x+2}{\sqrt{x}\left(x^2 + x + 1\right)^4}$$ Discuss the convergence of $\displaystyle\int_0^1f(x)\,\mathrm dx$ and $\displaystyle\int_1^{+\infty}f(x)\,\mathrm dx$. I encountered ...
1
vote
4answers
71 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
20
votes
1answer
659 views

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Another integral similar to my previous question: $$\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$$ Could you suggets how to evaluate ...
4
votes
1answer
178 views

Approximating an integral with another integral with finite limits

I came across the following integral in my work $$\int_{-\infty}^{\infty} \frac{\frac{1}{(1- \ \ 2 \pi j s \theta)^{m}}-1}{2\pi j s }\ e^{-2\pi j s\sigma^2}\ ds $$ Assuming $\theta,m,\sigma^2$ are ...
5
votes
1answer
136 views

How evaluate the following hard integrals?

Prove: $$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{\cos2x}{2\sin^2x}}}dx=\frac{\pi}{96}[{\pi^2}-6\ln^22]$$ And ...
5
votes
4answers
167 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
6
votes
2answers
111 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
3
votes
6answers
155 views

Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions

Let $$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$ How to show that it converges with no use of trigonometric functions? (trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be ...
7
votes
5answers
321 views

Evaluate $\int_0^1\frac{x^3 - x^2}{\ln x}\,\mathrm dx$?

How do I evaluate the following integral? $$\int_0^1\frac{x^3 - x^2}{\ln x }\,\mathrm dx$$
8
votes
2answers
119 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
5
votes
3answers
180 views

How to evaluate $\int_{0}^{\infty}\frac{(x^2-1)\ln{x}}{1+x^4}dx$?

How to evaluate the following integral $$I=\int_{0}^{\infty}\dfrac{(x^2-1)\ln{x}}{1+x^4}dx=\dfrac{\pi^2}{4\sqrt{2}}$$ without using residue or complex analysis methods?
11
votes
4answers
191 views

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

Nowadays I encounter an integral which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} ...
1
vote
0answers
27 views

$\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$

I was thinking about $\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$ The inspiration came from the following 3 integrals : Lemma If $f(x)$ is a bounded non-negative ...
0
votes
1answer
28 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
2
votes
2answers
79 views

Calculating a complex definite improper integral: $I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx$

Does anyone know how to find the value of this integral: $$I= \int_{0}^\infty x^{it}\,\mathrm{e}^{-ax}\, dx,$$ where $i=\sqrt{-1}$ and $t$, $a$ are real. Please give me a hint. Thank you.
3
votes
1answer
72 views

Show that $\int_0^r \frac{\mathrm{d}t}{\sqrt{r^2 - t^2}} $ is independent of $r$

I'm trying to show that the integral $$\int_0^r \frac{1}{\sqrt{r^2 - t^2}}\mathrm{d}t$$ is independent of $r$, without using trigonometric functions (namely, $t=\cos s$ and such). Can it be done? ...
5
votes
2answers
155 views

Evaluate $\int_0^{\pi/2}x\cot{(x)}\ln^4\cot\frac{x}{2}\,\mathrm dx$

How to evaluate the following integral ?: $$ \int_{0}^{\pi/2}x\cot\left(\, x\,\right)\ln^{4}\left[\,\cot\left(\,{x \over 2}\,\right)\,\right]\,{\rm d}x $$ It seems that evaluate to $$ {\pi \over ...
2
votes
1answer
79 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
1
vote
2answers
65 views

Does $\int_0^\infty |f'(x)| dx < \infty$ conclude $\lim_{x\to \infty} f(x)<\infty $

$f:[0,\infty) \to \mathbb R $ is $C^1$ and $$\int_0^\infty |f'(x)| dx < \infty$$ then can we prove that $\lim_{x\to \infty} f(x)$ exists and $$\lim_{x\to \infty} f(x)<\infty $$ My attempt: ...
1
vote
0answers
63 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
40
votes
1answer
1k views

Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi} $

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
20
votes
5answers
3k views

How to find the integral $\int_{0}^{\infty}\exp(- (ax+b/a))\,dx$?

How do I find $$\large\int_{0}^{\infty}e^{-\left(ax+\frac{b}{x}\right)}dx$$ where $a$ and $b$ are positive numbers? This is not a homework question. I will be quite happy if somebody can come up ...
6
votes
4answers
217 views

How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$

I found this nice result. Prove that $$\int_{0}^{\infty}\sin{x}\arctan\left({\frac{1}{x}}\right)\,\mathrm dx=\frac{\pi }{2} \left(\frac{e-1}e\right)$$ I tried some methods but I can't ...
0
votes
0answers
21 views

fredholm integral equation endpoint singularity

I am trying to solve numerically the following: $$ f(x)=\int_0^1\dfrac{1+tx}{(1-tx)^3}f(t)dt$$ Are there any quadrature rule or any other method to handle this singularity? Plz suggest any expansion ...
2
votes
0answers
25 views

$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
1
vote
1answer
21 views

Volume of a solid formed as vertical limit goes to infinity

Here's the question: The way I have it set up currently is as follows: $V = \pi \lim_{a \to \infty} \int_1^a (a-1)^2 - (\frac{1}{\sqrt{x^5}} - 1)^2$ But how do I go from here? And is the working ...
1
vote
1answer
59 views

Why is this integral $\int_{-\infty}^{+\infty} F(f(x)) - F(x) dx = 0$?

Let $a_n > 0$ and $b_n$ real. Let $f(x)= x - \sum_{i=0}^{\infty} \dfrac {a_n}{x+b_n}$ Now apparently for every function $F(x)$ : $$\int_{-\infty}^{+\infty} F(f(x)) - F(x) dx = 0$$ If the ...
23
votes
3answers
1k views

Is it OK to evaluate improper integrals this way?

Today in class we learned that when you have an improper integral like this one: $$\int_{-\infty}^\infty {f(x)} \: dx$$ you must split it before you do the limits (like so): $$\lim_{a \to \infty} ...
0
votes
0answers
47 views

How do we calculate naturally the following integral :

Do you know a natural method to calculate the following integrals: $$ I = \int_{\mathbb{P}^{1} (\mathbb{R})} \dfrac{1}{x} dx\quad\text{and}\quad J = \int_{\mathbb{P}^{1} (\mathbb{C})} \dfrac{1}{z} dz. ...
4
votes
1answer
176 views

Evaluation of $I_{a,b} = \int_{1}^{+\infty} \frac{\ \exp{(-at)}}{ 1-bt} \ \mathrm{d}t $

How to evaluate this integral: $$I_{a,b} = \int_{1}^{\infty} \frac{\ \exp\left(-a t\right)}{ 1-b t} \mathrm{d}t $$ where $a, b \in R^*_+$ ?
3
votes
1answer
143 views

Evaluating $\int_0^\infty \frac{\log t}{1+t^2}\,\mathrm dt$ using residues

I want to integrate $$\int_0^\infty \dfrac{\log t}{1+t^2}\,\mathrm dt$$ using the residue theorem. The poles are at $i,-i$. If the integral were from $-\infty$ to $\infty$, I would consider ...
17
votes
5answers
263 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
11
votes
4answers
270 views

Finding $\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$

How do we prove that $$I(m)=\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x=\frac{\log{m}}{m^2-1}$$ I see that $$I(m)=\frac{\partial}{\partial m} \int_{0}^{\pi/2} \arctan({m\tan x}) \ ...
3
votes
3answers
79 views

Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$

How to find the Cauchy principal value of the following integral $$\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$$ How to start this problem?
13
votes
2answers
445 views

A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
2
votes
1answer
60 views

Help integrating this improper integral [closed]

I'm having a lot of trouble integrating this, can anyone help? $$\int_0^\infty\frac1{2x^3+3x+1}\mathrm dx$$
4
votes
1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
1
vote
1answer
78 views

What is the derivation of an integral?

Given an intergral $$ \int_{a(t)}^{b(t)} f(t, x) dx $$ What is the fomula to of its derivation? $$ \frac{\partial}{\partial t} \int_{a(t)}^{b(t)} f(t, x) dx $$ Would someone please give me a link to ...
2
votes
1answer
36 views

Convergence with differential

suppose $f$ is differentialbe on $[0,\infty)$, and $f'(x)\geq 0$, $f(0)>0$. Furthermore, $$\int_0^ \infty \frac{dx}{f(x)+f'(x)}<\infty,$$ show that $$\int_0^\infty \frac{dx}{f(x)}<\infty.$$ ...
16
votes
3answers
763 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 ...
3
votes
2answers
90 views

The integral of $e^{-x^2}$ [duplicate]

How can I integrate this by parts? It seems to become recursive. I'm familiar with the classical solution, and cannot use that here due to the constraints of this class. Here's the integral (to ...
2
votes
2answers
56 views

Evaluation of an improper integral

I was trying earlier today to prove the convergence of an improper integral. In order to test my conclusion, I plugged it in WolframAlpha, getting a really beautiful answer: $$ ...