Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [improper-integrals]

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 $\infty$ or $-\infty$, or as both endpoints approach limits.

7
votes
2answers
97 views

What's wrong with this proof that $0 = 1$?

Let $$f_n(x)=\frac{1}{\sqrt{\pi n}}e^{-x^2/n}.$$ Note that $f_n(x)\to 0$ uniformly as $n\to\infty$. [Proof: $0\leq f_n(x)\leq\frac{1}{\sqrt{\pi n}}$; given any $\epsilon > 0$, let $M=\left\lceil\...
2
votes
2answers
35 views

Need help with an Improper integral

I need to evaluate $$\int_0^{2\pi}\frac{d\theta}{a+\sin^2\theta}.$$ I immediately noticed how this is an integral of the form $\int_0^{2\pi} f(\cos(\theta),\sin(\theta))d\theta$. I first tried to ...
0
votes
0answers
33 views

How to evaluate the following improper integral? [duplicate]

$\int_{0}^{\infty}\frac{1}{1+x^{2n}}dx = ? $ Using the Beta function and Euler's reflection formula the result of the integral ought to be $\frac{\frac{\pi}{2n}}{\sin{\frac{\pi}{2n}}}$ But I ...
0
votes
1answer
24 views

Monotonicity of improper integrals [on hold]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto f(x)$, and $x \mapsto g(x)$. If $f(x) \leq g(x)$ $\forall$ $x \in \mathbb{R}$, and $\int_{-\infty}^{\infty}...
10
votes
3answers
168 views

Evaluate the definite integral $\int^{\infty }_{0}\frac{x \,dx}{e^{x} -1}$ using contour integration

My friend and I have been trying weeks to evaluate the integral $$\int^{\infty }_{0}\frac{x \,dx}{e^{x} -1} .$$ We have together tried 23 contours, and all have failed. We already know how to ...
9
votes
1answer
137 views

How to compute this improper integral?

Let $n\geq1$ be an integer and let $$I_n=\int\limits_{0}^{\infty}\dfrac{\arctan x}{(1+x^2)^n} \,\mathrm dx$$ Prove that $$\sum\limits_{n=1}^{\infty}\dfrac{I_n}{n}=\dfrac{\pi^2}{6} \...
-1
votes
1answer
47 views

Improper integral 1 [on hold]

integration $\displaystyle\frac{x\tan^{-1}{x}}{\sqrt[3]{1+x^4}}$ on the interval 0 to infinity. Is the above improper integral convergent? Plz give the solution
0
votes
1answer
49 views

A doubt about an integration [closed]

Suppose I have, $\int_{-\epsilon}^{\epsilon} \frac{\delta(x)}{x}dx= ?$ Now If I think of its asymmetric integrand, then ans is $0.$, Otherwise, it is $1/0=\infty$. (Divergent!) Which ans I should ...
1
vote
1answer
31 views

Asymptotic behavior of Fresnel-like integral of an exponential [on hold]

Given the integral $$ I(t) = \int_0^t \mathrm{d}x \exp(-[\alpha \cos x + \beta \sin x]),\quad \alpha,\beta\in \mathbb{R}, $$ how can one obtain the asymptotic behavior for $t \to \infty$?
0
votes
0answers
40 views

Explore absolute and conditional convergence of the integral $\int_{1}^{+\infty} \frac{\sqrt[3]{x}\ln x \cos(x^{2})}{2\sqrt[6]{x}+1} dx$ [closed]

Explore absolute and conditional convergence of the integral $$ \int_{1}^{+\infty} \frac{\sqrt[3]{x}\ln x \cos(x^{2})}{2\sqrt[6]{x}+1} dx$$ I think firstly I need to substitute $x = t^{6}$.
0
votes
1answer
15 views

A convergent improper integral with $\int k\chi_{\{f>k\}}$ divergent

If $\int_0^\infty f(x) dx\leq\infty$ , we regard it as "convergent" , does there exist such an $f$, which satisfies $\int_0^\infty f(x) dx<\infty$ , but $\int_0^{\infty}k\chi_{E_k}(x)dx$ "divergent"...
0
votes
1answer
46 views

Proving the divergence of the integral $\int_0^\infty \frac{e^{-x}}{x} dx$ [on hold]

I suppose this is some kind of comparison test. But I couldn't find a function to compare to. Any help?
1
vote
2answers
28 views

Improper Integral Question (exponential integral)

I'm trying to prove that $\int_{0}^{\infty} x^{t} e^{-x} dx$ diverges for $t \leq -1.$ I figured I'd split it up into $\int_{0}^{1} x^{t} e^{-x} dx + \int_{1}^{\infty} x^{t} e^{-x} dx.$ If I can ...
0
votes
1answer
40 views

The integrals $\int_0^\infty \frac{\mathrm{d}x}{(a+bx^c)^d(e+fx^g)^h}$ and $\int_0^\infty \frac{x^e}{(a+bx^c)^d}\mathrm{d}x$

I am interested in improper integrals of rational functions. For example, I have found that $$\large{\int_0^\infty \frac{\mathrm{d}x}{(a+bx^c)^d}=\frac{\Gamma(\frac1c+1)\Gamma(d-\frac1c)}{\Gamma(d)a^{...
0
votes
1answer
29 views

Solve the limit of the following integral

Solve: $$ \lim_{n \rightarrow \infty} \int_{[0, \infty)} \frac{n \sin(\frac{x}{n})}{x(1+x^2)}\,\mathrm dm(x) $$ I am aware that the problem should be solved by using the Lebesgue's theorem of ...
-1
votes
1answer
36 views

Find all $\alpha\in\mathbb{R}$ such that integral converges.

Find all parameters $\alpha\in\mathbb{R}$ such that $$ \int_{1}^{+\infty}x\cdot\cos^{\alpha}\left(\frac{\pi}{2}\cdot\frac{x+1}{x+2}\right)\space dx$$ converges.
2
votes
2answers
89 views

Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$?

Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$? What have I tried: $$\int_0^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_0^1 \Big(\frac{\ln ...
-2
votes
0answers
60 views

How to solve the Integral $\int_{0}^{\infty} x^{-2}e^{-\left(a^2x^2+\frac{b^2}{x^2}\right)}dx$

How to solve the Integral $\int_{0}^{\infty} x^{-2}e^{-\left(a^2x^2+\frac{b^2}{x^2}\right)}dx$, if it is known that the value of $\int_{0}^{\infty} e^{-\left(a^2x^2+\frac{b^2}{x^2}\right)}dx=\frac{\...
4
votes
1answer
89 views

Evaluate $ \int\limits_{0}^{1} \frac{\arcsin{x}}{x} dx $

$$ \int\limits_{0}^{1} \frac{\arcsin{x}}{x} dx $$ The answer should be $1.0887...$ I have tried several substituions which didn't work out and also thought of some series expansions
0
votes
0answers
32 views

How to evaluate an improper integral $\int_{-3}^{-1}\frac{\sqrt{x^2 - 2x - 3}}{\sqrt[3]{(x^2 - 1)^4}}dx$

I only need to proof the convergence of this integral. $$\int_{-3}^{-1}\frac{\sqrt{x^2 - 2x - 3}}{\sqrt[3]{(x^2 - 1)^4}}dx$$
1
vote
0answers
11 views

Property of the remainder term in the Euler-Maclaurin formula for $\sum_{i=1}^n\log i$.

From https://en.wikipedia.org/wiki/Stirling's_approximation, I'm having trouble understanding $$R_{m,n}=\lim_{n\to\infty} R_{m,n}+O(n^{-m}).$$ I worked out $$R_{m,n}=\int_1^{n}\frac{P_m(x)}{mx^m}~ dx$$...
-1
votes
1answer
61 views

How to solve the integra $\int_{0}^{\infty} e^{-\left(a^2x^2+\frac{b^2}{x^2}\right)}dx$

How to solve $$\int_{0}^{\infty} e^{-\left(a^2x^2+\frac{b^2}{x^2}\right)}dx$$
2
votes
1answer
53 views

Explore absolute and conditional convergence of integral $\int_{1}^{+\infty} \frac{\sin \sqrt[3] x}{x-\ln x}\, dx$

Explore absolute and conditional convergence of the integral $$\int_{1}^{+\infty} \frac{\sin \sqrt[3] x}{x-\ln x}\, dx$$ My general ideas are the following: for absolute convergence I should use ...
2
votes
0answers
50 views

Find all parameters $\alpha \in \mathbb{R} $ such that $\int_{-1}^{0} \frac{2^{x+1} - 3^{x+1}}{\ln(\sin^{\alpha}(x+1)+1)} dx$ converges.

Find all parameters $\alpha \in \mathbb{R} $ such that $$ \int_{-1}^{0} \frac{2^{x+1} - 3^{x+1}}{\ln(\sin^{\alpha}(x+1)+1)} dx$$ converges. So I take an obvious substitution $t = x + 1$ which gives $$...
0
votes
2answers
28 views

Help solving an improper integral

The assignment is to solve: $$ \int_{0}^{\infty}f(x) dx $$ where $$f(x) = \frac{4}{(x+1)^2(x+3)} $$ . I did partial fraction on the indefinite integral of $f(x)$ and got: $$ f(x) = \frac{A}{x+1} + \...
1
vote
1answer
54 views

Proving $\int_0^{\infty}f(x) $ converges using Lagrange

Let $f:[0,∞) \to \Bbb R$ be a differentiable such that $f(x) > 0$ for every $x \in [0, \infty)$ and a positive function. Assume there exits an $0 < L < \infty$ such that $$\lim_{x\to \infty}\,...
0
votes
0answers
20 views

Integral involving upper incomplete gamma function, exponential and rational funcitons

Related to these questions here and here, I found a different form of the integrals, which result in $$f(x)=\frac{(-1)^n \,2^n}{\pi\,\lambda^{n+1}} \,\mathrm{e}^{-\lambda\,x}\int_{-\infty}^{\infty}\...
1
vote
3answers
46 views

Can we infer $\lim\limits_{x \to +\infty}f(x)=0$?

Assume that $f(x)$ is continuous over $[a,+\infty)$ where $a>0$, and $\displaystyle\int_{a}^{+\infty}\dfrac{f(x)}{x}{\rm d}x$ is convergent. Can we infer that $\lim\limits_{x \to +\infty}f(x)=0$? ...
1
vote
2answers
36 views

What is the definition of improper multi-variable integral with combination of unlimited integral area and infinity function value??

For example in $R^2$ with area $D=(a,\infty)\times(b,\infty)$ Function $f(x,y)$ is defined on $(x,y)\in D$ And $x=a,y=b$ are branch points, which means $\forall y\in(a,\infty)\Rightarrow\lim_{x\to a}...
1
vote
1answer
37 views

Function defined by integral - continuity, integrability, differentiability

Let $f: [0,+\infty) \to \mathbb{R}$ be Lebesgue integrable and define $$F(x) := \int^{+\infty}_0 e^{-tx}\cos(tx)f(t)dt $$ Determine if: a) F is defferentiable on $(0,+\infty)$ b) F is continuous ...
0
votes
3answers
60 views

Which functions $f(x)$ satisfies: $\int_0^\infty f(x) dx =0$ or $\int_{-\infty }^\infty f(x) dx = 0$?

I'm trying to find some $f(x) \neq 0$ that integrated from $0$ to $\infty$ or $-\infty$ to $\infty$ results in $0$. It can be $\int_0^\infty$ or $\int_{-\infty}^\infty$ doesn't matter since it goes to ...
2
votes
1answer
63 views

Can you prove that $\int_0^\infty \frac{|\sin\sqrt{qx}|-|\sin\sqrt{px}|}{x}dx = \frac{2}{\pi}\log\frac{q}{p}$? [closed]

This is a Frullani integral, but I am not sure if it converges. Anyway, I investigated it in my article on "fascinating integrals" (see here) if you are interested about how I came to that result.
1
vote
2answers
86 views

Inverse Fourier Transform of a half-integer Bessel function

Is there an analytical solution for the following inverse Fourier transform? $$f(x)=\frac{1}{\sqrt{2\pi}}\,\mathrm{j}^n \int_{-\infty}^\infty\frac{1}{\sqrt{k}\,(k^4-\lambda^4)}\mathrm{J}_{n+\frac{1}{...
2
votes
1answer
72 views

How to prove the integral converges?

Let $(\mathbf M'.\hat{\mathbf n})$ and $f(R,\theta)$ be a continuous function of $R$ and let $f(0,\theta)=0$. Then how shall we prove the following improper integral converges: $\displaystyle\lim \...
4
votes
3answers
70 views

Estimating a quotient of improper integrals

This was an extra credit question on my test on improper integrals. It was: Define $ \lfloor{x} \rfloor$ to be the greatest integer less than or equal to $x$, where $x$ is a real number. Calculate: ...
1
vote
2answers
32 views

Integral involving a generalized Laguerre Polynomial

I want to evaluate the following integral: $$ \int_0^\infty z^{1/2}e^{-a\space z}L_{m}^{1/2}(z)\space dz, $$ where $L_{m}^{1/2}$ is a generalized Laguerre polynomial. I found a certain indefinite ...
1
vote
3answers
64 views

Show that $\int\limits_2^{+\infty}\frac{\sin{x}}{x\ln{x}}\, \rm dx$ is conditionally convergent

The integral $\displaystyle\int_{2}^{+\infty}\frac{\sin{x}}{x\ln{x}}\,dx$ is conditionally convergent. I know that $\displaystyle\int_{2}^{+\infty}\frac{\sin{x}}{x}\, dx$ is conditionally convergent ...
4
votes
1answer
56 views

For f satisfying $\int_\mathbb{R}f(x)dx=M,\int_\mathbb{R}xf(x)dx=0,\int_\mathbb{R}x^2f(x)dx=E$, find $\inf\int_\mathbb{R}f(x)\log f(x) dx$

$f:\mathbb{R}\rightarrow[0,+\infty]$ $\int_\mathbb{R}f(x)dx=M,\int_\mathbb{R}xf(x)dx=0,\int_\mathbb{R}x^2f(x)dx=E$ Please evaluate $\inf\int_\mathbb{R}f(x)\log f(x) dx$ where $M$ and $E$ ...
1
vote
1answer
49 views

Is the $\int_0^x t^{-2/3}dt$ a continuous function of $x$?

Background: To be more general, if an improper integral of the form (where $f $ is unbounded at $0$)$$\int_0^bf(x)dx:=\lim_{\epsilon\rightarrow0^+}\int_\epsilon^bf(x)dx$$ converges, then can I ...
0
votes
1answer
46 views

On the Laplace Transform of $\frac{1}{1+\sin^2(x)}$

Are there any methods of evaluating this integral?$$I(s)=\int_0^\infty \frac{e^{-sx}}{1+\sin^2(x)}dx$$ I tried expanding the denominator into a geometric series to get $$I(s)=\sum_{n=0}^\infty (-1)^n\...
0
votes
1answer
22 views

Sequence of integrals converges to an improper integral proof?

Let $f: (0,1) \rightarrow \mathbb{R}$ be increasing on $(0,1)$. If the improper integral (of second kind) $\int_0^1 f(t)dt$ converges ($f$ is unbounded at $0$ and at $1$), then show that the sequence $...
0
votes
1answer
47 views

Convergence of $\int_0^{+\infty} \frac{\hat{f}(t)}{t} dt$ where $f$ is an odd integrable function on $\mathbb R$

In the process of proving that the fourier transform $L^1(\mathbb R) \rightarrow \mathcal C_0(\mathbb R)$ is not surjective, I am asked to show that given an odd integrable function $f$ on $\mathbb R$,...
1
vote
1answer
17 views

The condition for interchange path integral and integral over the real line

The question rises from a book I'm reading: Let $D$ be a domain, we have a function $g:D\subset \mathbb{C}\to \mathbb{C}$ defined by integration over a complex-valued fucntion $f$, where $f$ is entire ...
1
vote
0answers
29 views

Improper integral of a sinc function

I am trying to show that the sinc function $\frac{\sin(ax)}{x}$ behaves like a delta distribution when $\lim({a \to \infty})$. I can show that $$ \int_{-\infty}^{\infty} \frac{\sin(x)}{x}=\pi $$ ...
2
votes
2answers
29 views

Residue theorem /Integral

I want to calculate the following integral using residue theorem: $$\int_{-\infty}^{\infty} \frac{x^2}{x^4+1} $$ When I conisder the singularities, I get: $ \text{Rez}(f, z_k)=\frac{1}{4z_k}$ with $...
0
votes
1answer
46 views

An improper integral that diverges

I want to show that the integral \begin{align*} \int_1^{\infty} \frac{|\sin x|}{x} \text{ d}x \end{align*} diverges without sketching the function and obtain the divergence of the integral ...
2
votes
0answers
62 views

Why can’t improper integrals be defined directly using Riemann sums?

The standard way to define an improper integral of the form $\int_a^\infty f(t)dt$ is as follows. We first define the Riemann integral $\int_a^xf(t)dt$ for each $x>a$ in the standard way, i.e. ...
0
votes
1answer
50 views

A short way to determine the divergence of integral

The integral $$\int_0^1 \left| \frac{1}{x}\cos\left(\frac{1}{x}\right) \right| dx$$I can use the subsets in the form $$\left[ \frac{1}{(2k+7/3)\pi} , \frac{1}{(2k+5/3)\pi}\right]$$ where the cosine ...
2
votes
2answers
55 views

Bound for $\int_{0}^{t}e^{-xa^2}a(t-x)^{-b}\,dx$

I'm trying to estimate the following integral: $$\int_{0}^{t}e^{-xa^2}a(t-x)^{-b}\,dx$$ where constants $t,a>0$ and $0<b<1/2$. I want to get a bound for this integral. The preferred bound is ...
1
vote
1answer
60 views

Is there any other method of integration? besides the best known .

Is there any other method of integration besides the best known as: Substitution, Integration by parts, Trigonometric, Trigonometric Substitutions, Partial Fractions, Improper Integrals, and ...