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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.

-2
votes
0answers
24 views

Converts a single integral to the multiplication of two integrals [on hold]

How to write a single integral in the form of two integral multipliers? thank youenter image description here
13
votes
2answers
157 views

$\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}=\pi$ similar identities

I recently found that $$\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}=\pi$$ iff $$b^2-4ac=-4$$ I found it by integrating $$I=\int_{-\infty}^{\infty}\frac{\mathrm{d}x}{ax^2+bx+c}$$ If the ...
3
votes
1answer
81 views

How to prove $\int_{-\infty}^{+\infty} \frac{x^2}{\cosh(x)^2} dx = \frac{\pi^2}{6}$?

I found the integral in the Fermi gas theory. There is an approximate formula for specific integrals: $$\int_{-\infty}^{+\infty}F(\epsilon)\frac{\partial f(\epsilon)}{\partial \epsilon}d\epsilon\...
-1
votes
2answers
24 views

Why $\int_{(n-1)h}^{nh} f(x)dx≥f(nh)≥ \int_{nh}^{(n+1)h} f(x)dx$ is true is $f$ is decreasing? [on hold]

Why $$\int_{(n-1)h}^{nh} f(x)dx≥f(nh)≥ \int_{nh}^{(n+1)h} f(x)dx$$ is true is $f$ is decreasing?
0
votes
0answers
13 views

Is this a correct proof for convergence of Gamma Function

I have made the following proof for convergence of gamma function. Please tell me if it is correct. $\int_0^\infty e^{-x}x^{n-1}dx$ converges for all $n>0.$ Step I: $\int_0^1 e^{-x}x^{n-1}dx$ ...
0
votes
1answer
53 views

how to use $I(t)=\int_{0}^{\infty}\frac{e^{-t(x^2+1)}}{x^2+1}dx$ to calculate $\int_{0}^{\infty} e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$ [on hold]

how to use $$I(t)=\int_{0}^{\infty}\frac{e^{-t(x^2+1)}}{x^2+1}dx$$ to calculate $$\int_{0}^{\infty} e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$
3
votes
0answers
99 views

Convergence of $\int_0^\infty \sin(x^m)/x^n dx$

$$\int_0^\infty \frac{\sin (x^m)}{x^n}dx $$ Putting $x^m = t$ $$ \begin{align} \frac{1}{m}\int_0^\infty \frac{\sin t}{t^{(\frac{m+n-1}{m})}}dt \end{align} $$ By applying Dirichlet Test I've been ...
1
vote
1answer
32 views

$\int_0^\infty e^{-x}x^{n-1}dx$ is convergent for $n>0$

I have something to ask regarding convergence of gamma function. I have done the proof as below. Please tell me if it is correct. $\int_0^\infty e^{-x}x^{n-1}dx$ is convergent for $n>0$ Proof: ...
3
votes
0answers
53 views
+50

Closed form of an improper integral to solve the period of a dynamical system

This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period. In a special case, the integral is $$I=\int_{r_1}^{r_2}\frac{dr}{r\sqrt{\Phi^2(r,r_1)-1}}$...
1
vote
0answers
40 views

Closed from of $\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$?

I've been trying to find the general form of a certain group of integrals of the form$$I(a,n)=\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$$ I know that the real part of $I(a,2)$ can be calculated using ...
-1
votes
1answer
59 views

Convergence of $\int_0^\infty \frac{\sin x}{x}dx$ [on hold]

$$ \begin{align} \int_0^\infty \frac{\sin x}{x}dx \end{align} $$ Convergence at $x=0.$ Since, $\lim_{x\rightarrow 0}\frac{\sin x}{x} = 1$ so $0$ it is not a point of infinite discontinuity, hence the ...
1
vote
1answer
32 views

Uniform convergence of iterated improper integrals on $(0,\infty)$

I'm trying to get a better understanding of when it is permissible to swtich conditionally convergent improper integrals (when Fubini inapplicable) and I looked at a case where it works: $$\int_0^\...
3
votes
1answer
85 views

Need help computing the double integral $\int_{0}^{\infty} \int_{0}^{\infty} \frac{f(x + y)}{x + y} \mathop{dy} \mathop{dx}$

Need help computing the double integral $\int_{0}^{\infty} \int_{0}^{\infty} \frac{f(x + y)}{x + y} \mathop{dy} \mathop{dx}.$ I know that $\int_{0}^{\infty} f(u) \mathop{du}$ equals $1$. The entire ...
0
votes
2answers
43 views

Missing $i$ while evaluating $\int_{-\infty}^{\infty}\frac{e^{iz}}{(z^2+2z+2)^2}$ using residue theorem

Okay, first I'm a bit ashamed to ask because I already asked a question yesterday about a similar question (it's from far not the same integral though), but I'm missing an $i$ somewhere in the process,...
0
votes
0answers
20 views

What are all functions $f(x)$ that ensure $\int_{a}^{\infty} \frac{f(x)}{\sqrt{x^2-a^2}} \, \mathrm{d} x \le 0$ for all $a$ where $0 \le a \le \infty$

I'm looking to find a set of functions $f(x)$ such that members of the set satisfy the condition $$\int_{a}^{\infty} \frac{f(x)}{\sqrt{x^2-a^2}} \, \mathrm{d} x \le 0 \qquad \textrm{for all }0 \le a \...
0
votes
1answer
70 views

For which $p$, $q$ does the improper integral exist?

For which $p$, $q$ does the improper integral exist? $$ \text{(a)}\quad \int_0^\infty \frac{dx}{x^p+x^q} \qquad\quad\text{(b)}\quad \int_0^\infty x^p |x-1|^q \, dx $$ I tried for (a) to simplify ...
0
votes
3answers
63 views

Integral of $\int_0^\infty {x^2}{e^{-3x}}\,dx$

$$\int_0^\infty {x^2}{ e^{-3x}}\,dx$$ What I attempted was integration by parts twice, but I end with $-\frac{2}{27}$. That's obviously wrong, should be positive. I am also unsure whether or not $$\...
4
votes
2answers
97 views

solutions to $\int_{-\infty}^\infty \frac{1}{x^n+1}dx$ for even $n$

I was playing around with glasser's master theorem and integrals of the form $$\int_{-\infty}^\infty \frac{1}{x^n+1}dx$$ I observed that for positive, even values of n, the solution to the integral ...
0
votes
0answers
17 views

Suggestions for solving a complex triple integral

I am trying to solve the following integral: $\frac{1}{\sigma_1}\int_{-\infty}^{\infty}\phi\Big(\frac{y_{11}-\mu_{11}}{\sigma_1}\Big)\Big(BvN(h_{11},(y_{11}+\tau_1);\rho_3)-BvN(h_{12},(y_{11}+\tau_1);...
-1
votes
3answers
60 views

Showing that $\int_{0}^{\infty} u^{-\alpha} \sin(u) \, du >0$ for $0<\alpha<1$

Does anyone know how to show $\int_{0}^{\infty} u^{-\alpha} \sin(u) \, du >0$ for $0<\alpha<1$ (without explicitly having to calculate the exact value)?
0
votes
0answers
22 views

$\int_0^\infty\frac{x^m(1+x^n)}{1+x^p}$ is convergent [closed]

How to show that $$\int_0^\infty\frac{x^m(1+x^n)}{1+x^p}$$ is convergent if $p>1+m+n,m>0,n>0$ Please help me. I am clueless.
2
votes
1answer
56 views

how to show an improper integral is well defined?

Let $g$ be continuous on [0,1] and assume that $f$ is continuous on (0,1]. Assume further that for any $\epsilon\in (0,1)$, we have \begin{align*} \int_{\epsilon}^{1} f(s)\, ds=\int_{h(\epsilon)}^{1} ...
0
votes
0answers
11 views

How to show this distribution is proper?

This question is based on problem 14 from chapter 3 of Gelman et al.'s Bayesian Data Analysis. We have four data points $y_i$ with covariates $x_i$, $i=1,\cdots,4$. We use the model: $$y_i\sim \...
-1
votes
3answers
41 views

Prove that $\int_{0}^{\pi/2} \frac{\sin x}{x} dx$ converges [duplicate]

Prove that $\int_{0}^{\pi/2} \frac{\sin x}{x} dx$ converges. Shall I compare it with 1/x?
4
votes
1answer
23 views

Conditionally convergent integral

It is known that in a conditionally convergent series, the terms can be rearranged so as to output any desired value. Thus, a conditionally convergent series is said to be undefined. My question is ...
-4
votes
1answer
66 views

Evaluate the integral $\int _{0}^{4} \frac{dx}{\sqrt {4 - x}}$. [closed]

Evaluate the integral $\int _{0}^{4} \frac{dx}{\sqrt {4 - x}}$. Can anyone show me how to solve this please?
0
votes
1answer
38 views

test the integral $\int_{0}^{\infty} \frac {x}{3x^4 + 5x^2 +1}dx$ for convergence

test the integral $\int_{0}^{\infty} \frac {x}{3x^4 + 5x^2 +1}dx$ for convergence. My thought Can I compare it with 1/(3x^4)? Any hints for the solution are appreciated!
0
votes
2answers
39 views

if a finite limit $L = \lim_{x \rightarrow \infty} f(x) $ exists. Prove that $\exists M >0$ , $|f(x)| \leq M$ for all $x \in [a, +\infty)$

The question is given below: Let $f$ be a continuous function for $x \geq a,$ and suppose that a finite limit $L = \lim_{x \rightarrow \infty} f(x) $ exists. Prove that there exists $M >0$ such ...
1
vote
2answers
34 views

Evaluating $\int_{1}^\infty\frac{1}{t^a\sqrt{t^2-1}}dt$ for $a\geq 1$

I know that the this integral converges, but I can't show it. And, how can I proceed to calculate its value? $$\displaystyle \int_{1}^\infty \dfrac{1}{t^a\sqrt{t^2-1}} dt\qquad (a \geq 1)$$
3
votes
1answer
50 views

Finding $\int_0^\infty \frac{\arctan(p\cdot x)\cdot \arctan(q\cdot x)}{x^2} \text{d}t$

I am attempting to derive the value of the integral $$ I(p,q)= \int\limits_0^\infty \frac{\arctan(p\cdot x)\cdot \arctan(q\cdot x)}{x^2} \text{d}x $$ Differentiating the I w.r.t. p and then q gives ...
-2
votes
1answer
30 views

Determine whether $\int_0^\infty \sin(e^{\sin{x}})\, dx$ converges or diverges. [closed]

Determine whether $$\int_0^\infty \sin(e^{\sin{x}})\, dx$$ converges or diverges.
1
vote
2answers
82 views

Evaluating the value of $\int_0^{\infty} \frac {f(x)-f(2x)}{x} dx$

I want to evaluate the value of $\int_0^{\infty} \frac {f(x)-f(2x)}{x} dx$ where $f\in C([0,\infty])$ and $lim_{x\to \infty} f(x)=L$ I narrowed down the problem to showing $\int_0^\infty \frac{f(x)}{...
1
vote
0answers
43 views

Evaluating a multidimensional integral

I am solving a physics problem and ended up with the following integral: $\int_{-\infty}^{\infty} exp(-\beta\sum_{i=0}^{N}|\vec{r}_{i+1}-\vec{r}_{i}|^2)dr_{0}dr_{1}...dr_{N+1}$ where $\beta\,$ is ...
1
vote
0answers
40 views

Help with the specific improper integral for finding galaxy potential

I'm working on finding the potential of a razor-thin disk for a galaxy. I have encountered a specific improper integral for which I'm looking for an analytical expression. The integral is given below: ...
4
votes
2answers
104 views

How to prove $\int_0^\infty \frac{e^{-x-1/x}}{x \sqrt{x}} dx = \frac{\sqrt{\pi}}{e^2}$?

I went into this integration, $$\int_0^\infty \frac{e^{-x-1/x}}{x \sqrt{x}} dx = \frac{\sqrt{\pi}}{e^2}$$ Wolfram Online Integral Calculator gives the result above without process. May I know how to ...
8
votes
0answers
91 views

integrate $F(x)$: NO complex analysis, NO multivariable calculus

Suppose I have an elementary function $F(x)$ for which $\int_{-\infty}^\infty F(x) \, \text{d}x $ has an elementary value. Here 'elementary value' means anything generated by $0,1,+,-,\div,\times,\exp,...
2
votes
1answer
39 views

Good Book for methods of Convergence

First of all I'm interested in methods of convergence (for sums and integrals) so are there good books which tackle these methods one after the other and with examples that are not too trivial? For ...
2
votes
1answer
52 views

Does $(\alpha=\int_{-\infty}^{\infty} f(x)\, \mathrm{d}x)$ follow from $(\lim_{k\to \infty} \int_{a_k}^{b_k} f(x)\,\mathrm{d}x=\alpha)$?

Today I learned a bit about integrals and I'd like to know if that is true, or if there is a counterexample that disproves the following: When an $(\alpha\in \mathbb{R})$ exists, so that for all ...
2
votes
1answer
83 views

Calculation of some integrals

The next functions are defined: $$ f(y)=\frac{1}{1+e^{-2y}} \\ g_1(z)=\frac{1}{1+z^2},\quad g_2(z)=e^{-z^2},\quad g_3(z)=\frac{1}{cosh(z)},\quad g_4(z)=\frac{sin(z)}{z}$$ Is there a way to calculate ...
-4
votes
1answer
37 views

Im not able to find a counter example ? [closed]

Let $(f_n)$ be a sequence of continuous functions on $\mathbb R$. If $(f_n)$ converges to $f$ uniformly on $\mathbb R$ then $$\lim\limits_{n\to \infty}\int^{\infty}_{-\infty}f_n(x)dx \neq \int^{\...
0
votes
0answers
24 views

Testing convergence of improper integral with a variable a

I have trouble determining for which $a\in \mathbb{R}\;$ the following improper integral converges: $$\int_0^1\frac{\ln(x)}{x^a}dx$$ I have tried the following: $\left|\frac{\ln(x)}{x^a}\right|=\frac{-...
0
votes
1answer
42 views

Finding inverse Fourier transform of $\frac{1}{(1+iw)^2}$

Find the inverse Fourier transform of the function $$ \frac{1}{(1+iw)^2} $$ So I know the inverse is given by the integral $$ \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \frac{e^{iwx}}{(1+iw)^2}dw $...
2
votes
1answer
33 views

Show $ \int_ {-\infty} ^{\infty} (\arctan(x+a)) {{1}\over {\sqrt{2 \pi T}}} e^{-x^2 /2T} dx$ can assume any value in $(-\pi/2, \pi/2)$

I would like to show $$ \int_ {-\infty} ^{\infty} (\arctan(x+a)) {{1}\over {\sqrt{2 \pi T}}} e^{-x^2 /2T} dx$$ can assume any value in $(-\pi/2, \pi/2)$ where $T>0$ is fixed and $a$ may be any ...
2
votes
0answers
51 views

Improper integral with a parameter $\int_{0}^{\infty} e^{-cx^{2}}\sin(tx)dx$

I need to evaluate the following integral. $$\int_{0}^{\infty} e^{-cx^{2}}\sin(tx) ~dx$$ Here's what I've done so far. $$I(t) = \int_{0}^{\infty} e^{-cx^{2}}\sin(tx)~dx$$ Then differentiating under ...
1
vote
1answer
56 views

Limit of Improper integral from “0 to 0”

I have a problem which I think i'm close to solve, but I got stuck. The problem is as follows: Let f be continuous on [0,1] and let . Show that the following limit exists and find it: $$\lim_{\...
1
vote
1answer
55 views

Integrals $\int_{1}^{2} \frac{\mathrm dx}{\log x}$ and $\int_{0}^{+\infty} \left| \frac{\sin x}{x} \right| \mathrm dx$

I want to find out if the following integrals converge and if possible, find their values. $$(a) \int_{1}^{2} \frac{\mathrm dx}{\log x}$$ $$(b) \int_{0}^{+\infty} \left| \frac{\sin x}{x} \right| \...
0
votes
1answer
39 views

Does $\int_{-1}^1 \frac{f(x)g(x)}{\sqrt(1-x^2)} dx$ converge?

I was supposed to prove that $\int_{-1}^1 \frac{f(x)g(x)}{\sqrt(1-x^2)} dx$ converges. However, I am not sure if this actually converges because I was not given any information about $f(x)$ and $g(x)...
3
votes
0answers
52 views

Improper integral over the rationals

Question: Suppose that I wish to integrate a function over the natural numbers. How could I do this? Answer: Consider the definite integral $\int_a^bf(x)\ dx$. If we consider this as the 'area ...
1
vote
1answer
42 views

Finding all values $p$ for which $\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx$ converges

I've been stuck for a while with this exercise. Find all positive real values $p$ for which the integral $$\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx$$ converges. So far I've came up with ...
-1
votes
0answers
17 views

How to find definite intergral and calculate a value with FTC2

I had an exam today and one question kind of stumped me. It had two parts: 1) Find the derivitive, g'(x), of $\int_{cos(x)}^{sin(x)}\left(\frac{3}{v}\right)dv$ 2) g(pi/3) is defined by the definite ...