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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21 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} ...
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1answer
25 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
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4answers
153 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 ...
5
votes
2answers
88 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 ...
4
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1answer
109 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 ...
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6answers
152 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 ...
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0answers
24 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 ...
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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
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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} ...
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4answers
177 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)} ...
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2answers
64 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: ...
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0answers
62 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 ...
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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
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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 ...
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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 ...
23
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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} ...
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0answers
43 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. ...
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1answer
57 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 ...
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1answer
60 views

Help integrating this improper integral [closed]

I'm having a lot of trouble integrating this, can anyone help? $$\int_0^\infty\frac1{2x^3+3x+1}\mathrm dx$$
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5answers
253 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 ...
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4answers
211 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
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 ...
3
votes
2answers
89 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
178 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?
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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: $$ ...
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2answers
76 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 ...
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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
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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 $$\int_0^\infty \frac{dx}{f(x)}<\infty.$$ ...
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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 ...
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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 > ...
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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 $\frac{\sin \pi a x}{\pi x} = \frac{e^{i \pi a x} - e^{- i \pi a x}}{2i \pi x}$ ...
5
votes
2answers
152 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
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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 ...
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0answers
16 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 ...
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1answer
172 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 ...
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0answers
11 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 ...
2
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1answer
44 views

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

For which values of $p$ does the improper integral $$\int_0^\infty \frac{\log x}{1+x^p}\ dx$$ converge? I tried integration parts and various tricks, but it does not seems to work. Thanks
4
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0answers
86 views

Calculate the integral $\int_0^1\frac{\log^2(1+x^2)}{1+x}dx$ [closed]

Prove that: $$\int_0^1\frac{\log^2(1+x^2)}{1+x}dx=-\frac{\pi^2}{24}\ln2-\pi{G}+\frac{5}{2}\zeta(3)+\frac{2}{3}\ln^32$$
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0answers
33 views

Value of improper Integral

I need help in finding the value of the integral $$\displaystyle \int_0^\infty \left(\frac{x^2}{1+x}\right)^{n-1}e^{-tx}dx,$$ where $n$ is a positive integer and $t$ is a positive real number.
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3answers
28 views

Integral with absolute values?

How does one deal with improper integrals with absolute value bars? I need to show that a given function is a density function, which means I need to show that $\int_{-\infty}^\infty p(|x|) dx = 1$ ...
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?
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2answers
121 views

In what sense is $\int_{-\infty }^{\infty } \frac{x}{x^2+1} \, dx = \pi i$?

Suppose we want to give a meaning to the divergent integral $$I = \int_{-\infty }^{\infty } \frac{x}{x^2+1} \, dx,$$ perhaps in the sense of distributions or something (similarly to how $\int_{-\infty ...
2
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0answers
39 views

improper integrals in q-calculus

In quantum calculus is this equality possible for improper integrals? $\lim_{x\to\infty}\int_0^xf(t)d_qt=\int_0^\infty f(x)d_qx$
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1answer
91 views

Find $\int _{-\infty}^{\infty}\frac{1}{x^{12} + 1} dx$ elementarily [duplicate]

Ho to find the following integral $$\int_{-\infty}^{\infty}\frac{1}{x^{12} + 1} dx $$using parts by substitution, partial fractions, etc. but not Cauchy's residue theorem?
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2answers
38 views

Strange mistake when calculate improper integral

I want to calculate the following improper integral, so I do: $$ \int_0^{{\pi}/{2}} \cos^{2/3} x \sin^{-2/3} x\,dx = \int_0^{{\pi}/{2}} \frac{\cos x \cos^ {2/3} x}{\cos x \sin^{2/3}x}\,dx = ...
13
votes
3answers
278 views

Is there other methods to evaluate $\displaystyle\int_1^{\infty}\frac{\{x\}-\frac{1}{2}}{x}\ \mathrm dx$?

$$\displaystyle\int_1^{\infty}\frac{\{x\}-\frac{1}{2}}{x}\ \mathrm dx$$ where $\{x\}$ denotes the fractional part of x (for example- $\{3.141\}=0.141)$ I'm looking for alternative methods (maybe ...
3
votes
1answer
69 views

Evaluating $\int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} e^{-ixt} \,\mathrm dx$

$$\int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} e^{-ixt} \,\mathrm dx \quad \quad \quad \text{for }t>0$$ Use residue formula, which contour should I try?
7
votes
4answers
129 views

Proving $\int_0^1\frac{\log 2-\log\left({1+\sqrt{1-x^2}}\right)}{x}dx=\frac{\left(\pi^2-12\log^22\right)}{24}$

$$\int_0^1\frac{\log 2-\log\left({1+\sqrt{1-x^2}}\right)}{x}dx=\frac{\left(\pi^2-12\log^22\right)}{24}$$ At first, I think it can be calculated like the following one with differential method. ...
6
votes
5answers
107 views

Evaluating $\int_0^1\frac{x^{-a}-x^{a}}{1-x}\,\mathrm dx$

How to evaluate following integral $$\int_0^1\frac{x^{-a}-x^{a}}{1-x}\,\mathrm dx$$ I tried the feynman way $$\begin{align} I'(a)&=\int_0^1\frac{-x^{-a}\ln x-x^{a}\ln x}{1-x}\,\mathrm dx\\ ...