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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0
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1answer
18 views

improper integral - claim of convergences

Claim: Let $f:[a,b) \to R $ be a non-negative and continuous function. Suppose $\int_{a}^{b} e \ ^{f(t)} dt $ converge , than $\int_{a}^{b} f(t) \ ^ 7dt$ converge too. I think this claim is true. ...
8
votes
2answers
490 views

$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$ must be zero and it isn't

I'm trying to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$$ using residue and complex plane integration theory. Let $f(t):=\frac{\sin (t)}{t^4+1}$, $f(z):= \frac{\sin (...
0
votes
2answers
63 views

Maxwellian integral : is there a closed form?

$f_A(x,y)=\int_0^\infty du \frac{u \left(e^{-\frac{(u-x)^2}{2 A}}-e^{-\frac{(u+x)^2}{2 A}} \right)}{\sqrt{2 \pi } \sqrt{A} x \left(y^2+u^2\right)} $ is there a closed form? I was able to find ...
2
votes
1answer
42 views

Closed form for this integral (looks like Bessel)

I'm struggling to find a closed form for the following distribution (which is after all a Fourier Transform) written in integral form: $$I=\int_0^\infty\!\!\text{d}k\ \frac{ k }{\sqrt{k^2+m^2}}\sin(k ...
1
vote
1answer
44 views

Improper integral of rational function

I was computing some kind of marginal likelihood and came up with the following improper integral, $$\int_{-\infty}^{\infty}\frac{e^{kx}}{\prod_{i=1}^{m} (e^x+a_i)^{b_i}}dx$$ Or, $$\int_0^\infty\...
2
votes
6answers
946 views

Does the improper integral $\int_0^\infty\sin(x)\sin(x^2)\,\mathrm dx$ converge

Does the following improper integral converge? $$\lim_{B \to \infty}\int_0^B\sin(x)\sin(x^2)\,\mathrm dx$$
3
votes
4answers
167 views

How solve $\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$ [closed]

What is the value of the following integral? $$\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$$
2
votes
1answer
56 views

Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
1
vote
1answer
51 views

Can you prove the convergence of $ \int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx $?

Can you prove the following improper integral is convergent? $$ \int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx. $$
11
votes
3answers
199 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How can we calculate $$ \int_{0}^{\infty}{\sin\left(x\right)\sin\left(2x\right)\sin\left(3x\right)\ldots \sin\left(nx\right)\sin\left(n^{2}x\right) \over x^{n + 1}}\,\mathrm{d}x ? $$ I believe that ...
1
vote
1answer
54 views

Evaluating $\int_{s^{-1/n}}^{\infty}v^2\exp{\left[-\left(\frac{l}{v} + m v\right)^2\right]}dv$

I am trying to evaluate the following $$I = \int_0^s u^{-3n-1} \exp{\left[-\left(l u^n + \frac{m}{u^{n}}\right)^2\right]}\,du,$$ where $l, m$ and $n$ are positive constants. I tried to substitute $v ...
1
vote
1answer
37 views

Evaluating real integral by complex contour method

Please let me know where my mistake could be. I've verified the integral $$\int_{-\infty}^\infty \frac{dt}{(t^2+1)(t^2+4)}$$ to be equal to $\frac{\pi}{6}$ with a computer math system. However, I'm ...
2
votes
1answer
68 views

For which $\alpha$, $\beta$ does $\int\limits_1^{\infty} x^{\alpha} \cdot (\ln x)^\beta dx$ converge? [duplicate]

For which $\alpha$ and $\beta$ does the following integral converge ?: $$ \int_{1}^{\infty}x^{\alpha}\,\ln^{\beta}\left(x\right)\,\mathrm{d}x $$ Here is my analysis: I noticed that the function ...
0
votes
2answers
98 views

Integrate $I=\int_0^{\pi} \left(\cos(\theta)\right)^n \cos(p\theta) d\theta$

Is there any standard solution or way to solve the following integration $$I=\int_0^{\pi} \left(\cos(\theta)\right)^n \cos(p\theta) d\theta$$ where, $n=0, 1, 2,\dots$ and $p=0, 1, 2,\dots$ and $p>...
1
vote
1answer
70 views

Prove that $\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$ converges

Prove the convergence of $$\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$$ First I thought the integral does not converge because $$\int\limits_1^{\infty} -\frac{1}{x} \,\mathrm{d}x \...
1
vote
1answer
37 views

How to evaluate this inverse fourier transform integral

I am trying to do the following integral to evaluate an inverse fourier transform $$I = \frac{1}{2\pi}\int _{-\infty }^{\infty }k^{-i \xi } \left(c^{i \xi } - c^{1-i \xi}\right) \exp \left[- i m \xi ...
2
votes
2answers
44 views

For $\alpha>0$, $\int_{0}^{+\infty}\frac{t-\sin{t}}{t^\alpha}\,dt$ converges iff $\alpha\in (2,4)$

I can't prove that for $\alpha>0$, $I_\alpha=\int_{0}^{+\infty}\frac{t-\sin{t}}{t^\alpha}\,dt$ converges iff $\alpha\in (2,4)$. Here's my attempt: 1) An easy remark but important: as $\sin{t}\le t$...
2
votes
3answers
76 views

Gauss-Laguerre quadrature

I am trying to compute this integral: $$ \int_{0}^{\infty}\prod_{k = 1}^{d}\left(1 - \,\mathrm{e}^{-a_{k}\,t}\right) \,\mathrm{e}^{-t}\,\mathrm{d}t,\quad \mbox{where}\quad a_{k} > 0, \forall\ k. $$ ...
3
votes
1answer
25 views

Investigate convergence with cos/sin

Well I want to investigate the convergence of the following integrals(in the linked picture): $$\int_{1}^{\infty}\cos(x^t)dx\quad,\quad t\in \mathbb{R}$$ $$\int_{1}^{\infty}\sin(x^t)dx\quad,\quad t\in ...
1
vote
5answers
60 views

Improper Integral Exists Since a Limit Exists

I have read a solution which I didn't understand. $$ \mbox{Given the improper integral:}\quad \int_{0}^{1}{\,\mathrm{e}^x - \,\mathrm{e}^{-x} - 2x \over 2x^{2}\left(\,\mathrm{e}^x - \,\mathrm{e}^{-x}\,...
0
votes
0answers
23 views

Integral of Product of modified Bessel function, exponential functions and power function

I am trying to evaluate the definite integral 1 to obtain a closed-form solution, with z being the integration variable, and the other parameters are real positive constants. The integral can be ...
0
votes
0answers
15 views

$p$-power integral and $p$-series in higher dimensions

This seems like a basic question that should be addressed in a multivariable calculus course however I don't think I've ever confronted the issue until I became confused about a recent question here ...
0
votes
2answers
18 views

convergence of improper Integral..

I need help finding if the improper integral below converges. $$\int _{ 2 }^{ \infty }{ \frac { dx }{ \sqrt [ 3 ]{ 1-{ x }^{ 4 } } } } $$. we learnt at class: comparison test ratio test Thanks ...
-1
votes
1answer
21 views

Prove that $\int_{1}^{\infty} \cos(x)\cdot \frac{\sqrt{\ln(x)}\cdot x}{(x^2 + \sqrt[3]{\ln(x)})\cdot (x+5)}\,dx$ converges absolutely.

I'm attempting to prove that $\int_{1}^{\infty} \cos(x)\cdot \frac{\sqrt{\ln(x)}\cdot x}{(x^2 + \sqrt[3]{\ln(x)})\cdot (x+5)}\,dx$ converges absolutely. I've tried using the comparison test a few ...
1
vote
4answers
77 views

Proving that $\int_{1}^{\infty} \frac{\sin^{2}(3x)}{x} dx$ diverges

I have to prove that $\int_{1}^{\infty} \frac{\sin^{2}(3x)}{x}\,dx$ diverges. Can anyone give a hand? I'm totally stuck.
0
votes
1answer
26 views

Convergence of $\int_1^\infty\int_1^\infty(x+y)^{-a}dxdy$

My question is as follows: Find $a>0$ so that $I=\int_1^\infty\int_1^\infty(x+y)^{-a}dxdy$ converges. My attempt: Assume that $I$ converges. $I=\int_1^\infty x^{-a}\int_1^\infty(1+y/x)^{...
0
votes
1answer
62 views

Interchanging a limit and an integral

Prelude: Suppose I have the following integral $$\int^1_0 dt \frac{t}{1-t}$$ which is divergent. I want to see how the divergence manifests. I can see two approaches Rewrite the integral as $${\rm ...
2
votes
1answer
122 views

Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs

In my study of dynamical systems I was recently met with this system of ODEs: $ \dot{x}=\frac{\sinh{(y)}}{\cosh{(y)}+A\cos{(x)}} $ $ \dot{y}=\frac{A\sin{(x)}}{\cosh{(y)}+A\cos{(x)}} $ for a ...
0
votes
4answers
173 views

Evaluate $\int_{0}^{\infty} (-1)^{\lfloor x\rfloor}\cdot e^{-x} dx $ [closed]

I'm having trouble integrating the following: $$\int_{0}^{\infty} (-1)^{\lfloor x \rfloor}\cdot e^{-x} \, \mathrm{d}x $$ where $\lfloor x \rfloor$ denotes the floor of $x$. Can you help please?
0
votes
1answer
50 views

What is the domain for which this integral transform is defined?

Let $s=\sigma+it$ the complex variable where thus $i^2 =-1$, and $\sigma$ and $t$ are real numbers. Let $\mu(k)$ the Möbius function. It is possible determine the set of functions such that $$M \...
2
votes
4answers
93 views

Calculus Improper Integral Convergence; Which is right: Limits or Areas?

Could someone please explain to me the following doubt I have on improper integral: $$\int_{-\infty}^{\infty} \frac{1}{x} \ \mathrm{ dx}$$ I still think that since integrals signify areas that this ...
2
votes
1answer
52 views

Solving an improper integral contour integral, calculated via Wolfram but in need of analytic derivation possibly

In my studies of dynamical systems I have just encountered this supposedly tough looking improper integral, which is (not really relevant for my predicament) the Melnikov function, with the integral ...
1
vote
1answer
84 views

how to solve $ \int \frac{ e^{4cy^3+2by^2 + (a-3c)y - b}} {\sqrt{1-y^2}} dy $?

here $a,b,c$ are constants it can be solved as indefinite integral or a definite integral with limits [-1,1] or [0,1] MATLAB is not helping here
2
votes
1answer
60 views

Is this a finite integral?

A book on probability theory I am reading asserts the following: for $x, y \in \mathbb{R}^n, t \geq 0$, consider the function $h_t(x, y) = \frac{1}{t^{n/2}}e^{-\frac{r^2}{4t}}$, where $r = |x - y|$. ...
1
vote
3answers
63 views

Solving an Improper Integral: $\int_0^\infty r^2 e^{-a\cdot r}dr$

I have an improper integral as follows: $$\int_0^\infty r^2 e^{-a\cdot r}dr.$$ I try to evaluate it by parts and get $ [ -(\dfrac{r^2}{a} + \dfrac{2r}{a^2} + \dfrac{2}{a^3}) \cdot e^{-a \cdot ...
1
vote
1answer
45 views

Lebesgue integral of $\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}$ on an infinite cylinder

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell, and let $\boldsymbol{r}\in\mathbb{R}^3\setminus\overline{V}$ be any point external to $V$. I intuitively suppose that ...
1
vote
2answers
62 views

Convergence of improper integral: x^2*cos(e^x)

I am pretty sure the following converges, but I'm having trouble proving it. $$\int_0^\infty x^2\cos(e^x)$$ I tried breaking up the integral into pieces, where each piece corresponds to a period of $...
0
votes
0answers
64 views

Calculation of an integral with exponential and logarithmic functions

I am trying to do the following integral $$I = \int_{x}^{\infty}{ \frac{1}{\sqrt{u}} k^{\frac{2}{c}\ln u} \left(c \ln \left(\frac{k}{u}\right) - 2 c + 2 \ln^2 \left(\frac{k}{u}\right)\right) \exp \...
2
votes
1answer
43 views

Integration of an improper integral and the Cauchy principal value

I have been trying to evaluate the following integral: $$\int^{0}_{-\infty}e^{-i\omega t}dt$$ but I'm having trouble arriving at the correct result. My workings so far are as follows: $$\int^{0}_{-\...
6
votes
5answers
1k views

Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$

Compute improper integral : $\displaystyle I=\int\limits_{0}^{+\infty}\dfrac{\arctan \pi x-\arctan x}{x}dx$.
2
votes
1answer
62 views

Method for calculating integral of $e^{-2ix\pi\psi}/(1+x^2)$

I am seeking the method for calculating the following integral $$\int_{-\infty}^\infty\frac{e^{-2ix\pi\psi}}{1+x^2} dx $$ Ideas I have are: 1) substition (however which one?) 2) integration by ...
0
votes
0answers
27 views

How to calculate the integral of a fourier trasfom

I have to calculate this integral : $\int_{-\infty}^{+\infty} \hat G(\omega)e^{i\frac{\pi}{2}\omega}d\omega$ ($\hat G(\omega)$ is the Fourier trasform) with: $G:x\in \Bbb{R}\to \begin{cases} g(x), ...
-1
votes
2answers
69 views

Integral involving exponential and power

I have bumped into the following integral, which Mathematica is apparently not able to solve. I have tried a couple of change of variables and a series expansion for $x$ close to $1$, but without much ...
0
votes
1answer
46 views

Determining if an integral is absolutely convergent

Any idea why the following integral is conditionally convergent and not absolutely convergent? $$\int_{1}^{\infty} \frac{\sin(3x)\cdot \arctan(x)}{\sqrt{x}} dx $$
8
votes
2answers
188 views

Prove that $\int_{0}^{1}{(1-x)(x-3)\over 1+x^2}\cdot{dx\over \ln{x}}={\ln{8\over \Gamma^4(3/4)}}$

Prove $$I=\int_{0}^{1}{(1-x)(x-3)\over 1+x^2}\cdot{dx\over \ln{x}}=\color{blue}{\ln{8\over \Gamma^4(3/4)}}\tag1$$ $(1-x)(x-3)=-x^2+4x-3$ $${1\over 1+x^2}=\sum_{n=0}^{\infty}(-1)^nx^{2n}\tag2$$ ...
15
votes
1answer
701 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} \frac{t_i^2}{g+\lambda_i^{\psi}}...
5
votes
1answer
90 views

Show that $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$

Show that the improper integral $\int_0^1 \left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm dx=\alpha \ln\alpha$, for $\alpha\in(0,1)$. This is ...
2
votes
1answer
44 views

Ways to justify this interchange of summation and integration

In evaluating this integral: $$\int_0^\infty \frac{\Im{\left(e^{e^{ix}} \right)}}{x}\text{d}x$$ My means of evaluation was to expand the numerator of the integrand as a fourier series (a.k.a. Taylor ...
117
votes
21answers
31k views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_{0}^{\infty} \frac{\sin{x}}{x} \, dx = \frac{\pi}{2}$$ Well, can ...
5
votes
2answers
123 views

Evaluate $\int_0^\infty \frac{dx}{x^2+2ax+b}$

For $a^2<b$, is there an identity of evaluating the following integral? $$\int_0^\infty \frac{dx}{x^2+2ax+b}$$ What about: $$\int_0^\infty \frac{dx}{(x^2+2ax+b)^2}$$ My attempt is ...