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.

learn more… | top users | synonyms

0
votes
1answer
20 views

Finding the residue of the improper integral $\frac{1}{z^4+4}$

$$f(z) = \frac{1}{z^4+4}$$ the roots of this are: $z^2=\pm i\sqrt{2} \implies z=\pm\sqrt{i\sqrt{2}}$ and $z=\pm i\sqrt{i\sqrt{2}}$ i.e. $$f(z) = \frac{1}{(z\pm\sqrt{i\sqrt{2}})(\pm ...
2
votes
3answers
61 views

If $\lim_{x\to\infty}\int_0^x a(t)\,dt=L$, is it true that $\lim_{x\to\infty}a(x)=0?$

Let $s(x)=\int_0^xa(t)\,dt$. I wonder: If $\displaystyle \lim_{x\to\infty}s(x)=L$ is it true that $\displaystyle\lim_{x\to\infty}a(x)=0?$
10
votes
3answers
209 views

What are other methods to evaluate $\displaystyle\int_0^1 \sqrt{-\ln x} \ \mathrm dx$

$$\int_0^1 \sqrt{-\ln x} dx$$ I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral. $$y=-\ln x$$ $$\bbox[8pt, border:1pt solid ...
0
votes
0answers
41 views

Contour integration when pole is outside the contour

Here they are using the pole OUTSIDE the contour? I thought this was illegal according to the residue theorem or we are not supposed to do contour integration with poles outside the contour itself.
2
votes
1answer
46 views

How many poles have to be inside the contour?

If we consider $$\int_{0}^{\infty} \frac{dx}{1+x^2}$$ Using complex contour integration only. We choose a contour in the TOP HALF plane. From the poles $z = \pm i$ only, the pole: $z=i$ is ...
1
vote
1answer
22 views

Limits for each part of the improper integral, why?

My handbooks "says" that a limit must be calculated for each part of the improper integral. When using the same limit for all parts, it is called the Cauchy Principal Value. My question is, what the ...
3
votes
2answers
73 views

How do I evaluate the following integral $\int_{-\infty}^{\infty} e^{-\sigma^2 x^2/2}\; \mathrm dx$? [duplicate]

How do I evaluate the following integral $$\int_{-\infty}^{\infty} \exp\left(-\frac{\sigma^2 x^2}{2}\right) \mathrm dx\;?$$ How is it even possible to find an antiderivative? The integral is ...
5
votes
1answer
65 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 $f$ ...
0
votes
1answer
60 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
26 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
90 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 ...
3
votes
2answers
140 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} ...
1
vote
1answer
79 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: ...
5
votes
4answers
190 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 ...
7
votes
2answers
122 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 ...
5
votes
1answer
148 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 ...
3
votes
6answers
156 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 ...
1
vote
0answers
32 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
1answer
82 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} ...
11
votes
4answers
203 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
2answers
67 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
64 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 ...
0
votes
0answers
22 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
27 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 ...
27
votes
4answers
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
48 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. ...
2
votes
1answer
68 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 ...
2
votes
1answer
60 views

Help integrating this improper integral [closed]

I'm having a lot of trouble integrating this, can anyone help? $$\int_0^\infty\frac1{2x^3+3x+1}\mathrm dx$$
17
votes
5answers
272 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 ...
6
votes
4answers
221 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 ...
4
votes
1answer
28 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 ...
3
votes
2answers
91 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 ...
5
votes
3answers
184 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?
2
votes
2answers
59 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: $$ ...
1
vote
2answers
80 views

Is it possible to evaluate $\int_0^1 \sin(\frac{1}{t})\,dt\,$?

I was wandering if it possible to evaluate the value of the following improper integral: $$ \int_0^1 \sin\left(\frac{1}{t}\right)\,dt $$ It is convergent since $\displaystyle\int_0^1 ...
6
votes
3answers
152 views

Evaluate $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$

Evaluate $$\displaystyle\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$$ How do I evaluate this integral? I know that the result is $0$, but I don't know how to obtain this. Wolfram|Alpha ...
0
votes
2answers
26 views

The product of two nonnegative, improperly integrable functions is also improperly integrable.

True or False: The product of two nonnegative, improperly integrable functions is also improperly integrable. I was given both the problem and the proof that may or may not be true. I think the ...
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 $$\int_0^\infty \frac{dx}{f(x)}<\infty.$$ ...
0
votes
1answer
29 views

An integral yielding dirac delta (edited)

Is the equation below true for $ x>0$ ? $$\lim_{x\rightarrow0}\left(\int_0^\infty \exp(-Rx)\cos(R(y-t)) \, dR\right)=\pi \delta(t-y)$$ Actually i don't understand why the multiplier $\pi$ exists ...
4
votes
0answers
29 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
0
votes
1answer
20 views

Fourier transform of a function with sine [duplicate]

I don't know how to compute the Fourier tranform of this function: $f(x) = \frac{\sin \pi a x}{\pi x}$ I know that $\frac{\sin \pi a x}{\pi x} = \frac{e^{i \pi a x} - e^{- i \pi a x}}{2i \pi x}$ ...
4
votes
2answers
165 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
0answers
36 views

Evaluating the integral $\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$

I'm searching for a way to evaluate the following integral: $$\int_0^a x^{v/2} e^{-\alpha x} J_v(2\beta\sqrt{x}) dx$$ where $J_v(x)$ are the Bessel-functions, and $v \in \mathbb{N}, (a,\beta) \in ...
2
votes
1answer
37 views

Integral of a gaussian over a slice of the plane

I need to evaluate the following $n$ real integrals: $$\int_{\frac{\pi}{2}-\frac{\pi}{n}}^{\frac{\pi}{2}+\frac{\pi}{n}}\int_0^\infty\frac{1}{\pi \sigma^2}e^{-\frac{|re^{i\theta}-i|^2}{\sigma^2}} \ dr ...
0
votes
0answers
17 views

Exercise: Integral with spherical and cylindrical coordinates

I have to solve two exercises with spherical and integrals coordinates. I have done this stuff a lot of time ago and I do not remember very well how does it work. I am having particular problems with ...
4
votes
1answer
179 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 ...
0
votes
0answers
12 views

Multiple integral of iterated kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel $$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$ should be calculated. However, it is not ...