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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3
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1answer
38 views

Riemann integral and Lebesgue integral (in real analysis Folland)

The following is from p.57(Real Analysis, by Folland) My question is following: Dominated convergence theorem should roughly be: $f_n \rightarrow f, |f_n|\leq g$ a.e., then $\int f = ...
4
votes
1answer
23 views

Show $\int_{s}^{\infty} f(x)dx = \mathcal{L} \{\frac{F(t)}{t}\}$ given $f(x) = \int_{0}^{\infty} e^{-xt}F(t)dt$

I'm trying to derive this to show that $$\int_{0}^{\infty} f(x)dx = \int_{0}^{\infty} \frac{F(t)}{t} dt$$ and use that to prove $$\int_{0}^{\infty} \frac{\sin t}{t} = \frac{\pi}{2}$$ How do I go ...
1
vote
0answers
31 views

Compute $\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $

Compute the integral $$\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $$ I assume contour integration is the easier way ?
3
votes
2answers
92 views

Modified Leibnitz integral: $\lim\limits_{a \to\infty}\frac1a\int _0^\infty\frac{(x^2+ax+1)\arctan(\frac{1}{x})}{1+x^4}dx=?$

$\lim\limits_{a \to \infty} \frac{1}{a} \int _0^\infty\frac{(x^2+ax+1)\arctan(\frac{1}{x})}{1+x^4}dx $ ,where $a$ is a parameter. ATTEMPT:- Let $I(a)=\frac{1}{a} \int _0^\infty ...
0
votes
0answers
25 views

Improper integral of trigonometric function

Trying to find the following improper integral's divergence, which I've split into two integrals: $\int_{0}^{\infty} \frac{\sin{x}}{x^2} dx = \int_{0}^{1} \frac{\sin{x}}{x^2} dx + \int_{1}^{\infty} ...
0
votes
1answer
28 views

Am I correctly evaluating the improper integral $\int_9^{\infty} \frac{1}{\sqrt{x^5}} dx$

$$\int_9^{\infty} \frac{1}{\sqrt{x^5}} dx$$ $$u=\sqrt{x^5}\implies u=x^{5/2} \implies x=u^{2/5}\implies dx=\frac{2}{5u^{3/5}}$$ $$\int_{9^{5/2}}^{\infty} \frac{2}{5u^{3/5}u} du$$ ...
1
vote
0answers
68 views

Does $\frac 1 n \int_0^n \frac {g(x)}{h(x)}\,dx\to 0$ if $\int_0^\infty g(x)\,dx$ converges, $\int_0^\infty h(x)\,dx$ diverges and

$h(x)$ is monotone positive on $[0,\infty)$? It seems like it should be the case since $g$ must go down to $0$ faster than $h$, but I can't formalize the argument since $\frac g h$ can oscillate. ...
7
votes
1answer
217 views

How to evaluate $\int_0^{\pi /2}\frac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos u)})^2}du$?

I want to find the value of $$\int_0^{\pi /2}\dfrac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos u)})^2}du.$$ Let $v=\frac{\pi}{2}-u$, then $$\int_0^{\pi /2}\dfrac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos ...
0
votes
1answer
80 views

Evaluate integral $\int_{-1}^{1} x^2 \exp(\frac{1}{x^2-1}) dx$

How to evaluate this integral? $\int_{-1}^{1} x^2 \exp(\frac{1}{x^2-1}) dx$ If possible, I want any solution in elementary/non-elementary functions. Many thanks.
2
votes
1answer
51 views

Evaluate $\int_{-1}^{1}\mathop{dx}\frac{1}{(x^2+a^2)^k}$?

My original question was what is the condition for the following integral to converge: $$\int_{-1}^{1}\mathop{dx}\left[\frac{1}{x^{2k}}-\frac{1}{(x^2+a^2)^k}\right].$$ I know $k>0$. By motivation ...
2
votes
1answer
46 views

Why is this change of variable contradictory?

I have a following integral $$\mathcal{I}=\int_0^\infty \mathop{dx}x^{-m}\frac{j_2(ax)}{(ax)^2},$$ where $j_2(x)$ is the spherical bessel function, $a>0$ is a real parameter and $m>1$. You do ...
5
votes
0answers
89 views

Recurrence for $\int \left(\frac{\sin x}{x}\right)^n \, \mathrm{d}x$ [duplicate]

I was playing around and wanted to consider the integral of $$I_n = \int_0^{\infty} \left(\frac{\sin x}{x}\right)^n \, \mathrm{d}x$$ using parts with $u = \sin^n x \implies \mathrm{d}u = n\cos x ...
0
votes
4answers
177 views

How to compute $\int_0^{+\infty} \frac{dt}{1+t^4} = \frac{\pi}{2\sqrt 2}.$

How to compute $$\int_0^{+\infty} \frac{dt}{1+t^4} = \frac{\pi}{2\sqrt 2}.$$ I'm interested in more ways of computing this integral. There is always the straight forward method to ...
0
votes
1answer
54 views

Munkres' definition of the extended integral

After defining the Riemann integral over bounded subsets of $\mathbb{R}^n$, Munkres' Analysis on Manifolds defines the improper integral as follows: Definition. Let $A$ be an open set in $R^n$; ...
0
votes
0answers
40 views

Prove that $\lim_{x\to\infty} f(x) = 0$ if $\lim_{x\to\infty}( f(x)+ \int_{0}^{x} f(x) dx)$ exists [duplicate]

Let f be a real valued continuous function on $[0,\infty)$ such that $\lim_{x\to\infty}( f(x)+ \int_{0}^{x} f(x) dx)$ exists. Prove that $\lim_{x\to\infty} f(x) = 0 $. If f is non negative function ...
2
votes
1answer
21 views

Divergence of a sequence involving logarithmic integrals

Let $\displaystyle a_n=\int_0^n\int_0^{\frac{1}{n}}\int_0^n\int_0^{\frac{1}{n}}\log^2\Big[(x-t)^2+(y-s)^2\Big]dx\,dy\,dt\,ds,$ for $n=1,2,\cdots$. How to show that ...
1
vote
1answer
46 views

$\int_0^1\frac{1}{r}\frac{1}{\left[\log\left(1+\frac{1}{r}\right)\right]^n}dr$ finite?

Does someone have a hint for me why the integral $\int_0^1\frac{1}{r}\frac{1}{\left[\log\left(1+\frac{1}{r}\right)\right]^n}dr$ is finite? $n$ is a natural number greater than $1$.
0
votes
1answer
17 views

Determing why the Integral comparison test cannot be used

Hey guys I have this homework problem that I am having difficulty explaining. The question is why the integral comparison test cannot be used. The Series is: ...
1
vote
1answer
36 views

To test the convergence of $\int_0^{1} \frac{x^p \log x}{1+ x^2} dx$

To test the convergence of the improper integral: $$\int_0^{1} \frac{x^p \log x}{1+ x^2} dx$$ Here we see that $0$ is the point of infinite discontinuty for $p<0$. Let $f(x) = \frac{x^p \log ...
2
votes
1answer
69 views

Proving $\int_0^{\infty} \frac {x^{m-1} - x^{n-1}}{1-x} dx$ and $\int_0^{\infty} x^m (\log x)^m dx$

Prove that: (a) $\int_0^{\infty} \frac {x^{m-1} - x^{n-1}}{1-x} dx$ is convergent if $0<m<1$ and $0<n<1$; (b) $\int_0^{\infty} x^m (\log x)^n dx$ is convergent if $m< -1$; Getting no ...
0
votes
1answer
24 views

About the the convergence of the improper integrals:

About the the convergence of the improper integrals: 1: $\int_0^{\pi/2} \frac{1}{e^x - \cos x} dx$ 2: $\int_0^{\pi} \frac{1}{\cos \alpha - \cos x} dx, 0 \leq \alpha \leq \pi.$ In the first problem ...
-1
votes
1answer
74 views

how do I find integrals where one limit tends to infinity?

As in this question, where putting in the value of infinity makes it unsolvable. $$\int_0^\infty \frac{1}{\sqrt{x} (x+1)} \, dx$$ so I wrote this integral as an integral with limits 0 to t where t ...
1
vote
1answer
29 views

To examine the convergence of the improper integrals

To examine the convergence of the improper integrals: 1: $\int_0^{\infty} \frac1{x \log x}dx$ 2: $\int_0^{\infty} \frac1{(x +sin^2x) \log x}dx$ In both the cases we see that $0$ and $1$ are point ...
0
votes
1answer
47 views

Investigate absolute convergence of the integral $\int_0^\infty x^2\cos e^x\,dx$

I am studying absolute convergence of improper integral over $\left[0,+\infty\right)$ $$\int_0^\infty\!x^2\ \cos(e^x)\ dx$$ And I used the substitution $t=e^x$, I produce the improper integral ...
1
vote
1answer
78 views

Improper integral substitution hint

i try solve this improper integral $$\int_0^\infty x^p\sin x^q \ dx$$ I try to compare it with $\displaystyle \int_0^\infty\ \frac{1}{x^p}\ dx$ But I don't know what do when $x\rightarrow\ \infty$ in ...
0
votes
0answers
20 views

Improper integration limits

Suppose I have a function $f(x)$ such that $\int\limits_{-\infty}^{\infty}f(x) dx = c$. Also, $f$ is even. Is the following equation correct? $$\int\limits_{a}^{b}f(x)dx = ...
2
votes
2answers
31 views

If $\int_1^\infty x^{-p} dx\,$ exists, then does $\int_1^\infty x^{-q} dx\,$ exist, where $q > p$?

If $\int_1^\infty x^{-p} dx\,$ exists, then does $\int_1^\infty x^{-q} dx\,$ exist, where $q > p$? My initial assumption is that the answer was true. Because if p is an arbitrary number 5, then ...
1
vote
1answer
27 views

If f is continuous and 0 < f(x) < g(x) on the interval [0, ∞) and $\int_0^\infty g(x) dx = M < \infty$ then $\int_0^\infty f(x) dx$ exists?

If f is continuous and $0 < f(x) < g(x)$ on the interval $[0, ∞)$ and $\int_0^\infty g(x) dx = M < \infty$ then $\int_0^\infty f(x) dx$ exists. True or False, and why? I'm not sure what the ...
1
vote
2answers
37 views

principal value of improper integrals

How can I find the principal value of the following? $$PV\int_{-\infty}^\infty \frac{e^{ix}}{x(x^2+x+1)} dx$$ I'm able to evaluate integrals which have trig identities in them or just polynomials ...
3
votes
1answer
46 views

Integrating $f(x,y)= \frac{1}{2 \pi (t-s)s} e^{- \frac{(y-x)^2}{2(t-s)}- \frac{x^2}{2s}}$

I want to integrate the function $$f(x,y)= \frac{1}{2 \pi (t-s)s} e^{- \frac{(y-x)^2}{2(t-s)}- \frac{x^2}{2s}}$$ on $[0,\infty)^2$ or in other words. I am looking for $$\int_{0}^{\infty} ...
2
votes
0answers
21 views

Singularity of an oscillatory integral

Given $x\in\mathbb{R}^3\setminus \{0\}$, consider the following integral: $$I(x):=\int_{\mathbb{R}^3}\frac{e^{-i|x-y|^2}}{|y|} \, dy$$ Now $I(x)$ diverges as $x$ approaches to $0$, and it seems to me ...
2
votes
1answer
75 views

How do I evaluate the following Integral

The integral is $$\int_{0}^{\infty}dx\frac{1}{\sqrt{1+a(1+x^2)^m+b(1+x^2)^{m-2}x^2}},$$ where $m, a$ and $b$ are real numbers such that the integral is definitely convergent. Any ideas on how to solve ...
-1
votes
1answer
34 views

If $f$ is continuous on [0,$∞$) and $\lim_{x\to\infty} f(x)= 0$, then prove that $\int_{0}^{∞} f'(x) dx = -f(0).$ [closed]

If $f$ is continuous on [0,$∞$) and $\lim_{x\to\infty} f(x)= 0$, then prove that $$\int_{0}^{∞} f'(x) dx = -f(0).$$ Any hints?
2
votes
1answer
46 views

Integral involving an exponential and logarithmic function

I am trying to find the expected value of a probability density function. Solving the integral of the function times its random variable with integration by parts, I arrive at the following integrals ...
2
votes
0answers
28 views

Solving integral $\int_{-\infty}^{\infty}\frac{e^{-(2\pi-i2\pi t) f}}{(3+i2\pi f)^2}df$

If Fourier transform of $h(t)$ is $H(f)=\dfrac{2}{(3+i2\pi f)^2}$ and Fourier transform of $x(t)$ is $X(f)=e^{-2\pi f}$, then determine the convolution of $h(t)*x(t)$. From the theorem ...
1
vote
2answers
43 views

Determining convergence of improper integrals including $ \int_{0} ^ {1} \frac{\ln\left(1+e^x\right)-x}{x^2}\text{d}x $

Will you please help me figure out whether the following improper integrals converge or not? $$ \int _ {0} ^ {\infty} \frac{x^2}{2^x}\text{d}x $$ $$ \int_{0} ^ {1} ...
0
votes
0answers
15 views

Determine integral range in function of 2 random variables. [duplicate]

X and Y are independent uniform r.vs in the common interval (0,1). Determine where Z = X + Y. There are 2 ranges of z: $0 < z < 1$ and $1<z<2$. The diagrams given for both ranges ...
1
vote
1answer
32 views

Integral of Dirac Delta Function

If the Dirac Delta function has an infinitely tall "spike" at $x=0$, then how is it that its integral over the entire real number line is one, instead of infinity?
3
votes
2answers
104 views

How do I evaluate the following integral $\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m}$?

I am interested in the following integral $$\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m},$$ given that $m>n/2$ (this is just what I wrote so that the integral converges. If this is not ...
2
votes
2answers
68 views

Other ways to compute this integral?

The following (improper) integral comes up in exercise 2.27 in Folland (see this other question): $$I = \int_0^\infty \frac{a}{e^{ax}-1} - \frac{b}{e^{bx}-1}\,dx.$$ I computed it as follows. An ...
0
votes
4answers
73 views

To show that: $\int_{-1}^{1} \frac{1}{(1+x)^{1/3}\ (1-x)^{2/3}} dx = \frac{2\pi}{\sqrt3}$

To show that: $$\int_{-1}^{1} \frac{1}{(1+x)^{1/3}\ (1-x)^{2/3}} dx = \frac{2\pi}{\sqrt3}$$...............(A) We see that $\frac{2\pi}{\sqrt3}$ can be written as $$\frac{\pi}{\sqrt3 /2} = ...
0
votes
3answers
40 views

Improper integration of a general function

I am trying to find some closed form answer for the integral $$\int_0^{\infty}\frac{x^n}{(x^2+1)^n}dx,2\leq n$$ I am not sure if a closed form exists and I have been trying this integral for hours. ...
4
votes
3answers
131 views

Proof for divergence of $\int_1^\infty \cos(x^\frac{3}{4})$

As the title says, I am unable to find a proper proof for the divergence of $$\int_1^\infty \cos(x^{3/4})$$ As its is not positive, I can't use any of the divergence tests, nor compare it to a sum. ...
1
vote
1answer
66 views

Evaluating $~\int_0^\pi\frac{\sin^{m-1}x}{(2+\cos x)^m}~dx$

How to find the following definite integral: $$\int_0^{\pi}\frac{\sin^{m-1}x}{(2+\cos x)^m}dx$$ I have done up to: $$\int_0^{\pi}\frac{\sin^{m-1}x}{(2+\cos ...
0
votes
3answers
35 views

Evaluate the integral (improper integration)

Evaluate $\int\frac{x^4+x+1}{x^3+x}\,dx$. I need help doing this problem, I started to do long division but I don't know how to because the coefficients don't match. Thank you
1
vote
1answer
39 views

Riemann improper integral problem (Introduction to the theory of integration)

For this question I am aware I can integrate it normally and get a solution, but is there a specific way required for the reimann improper integral? Do I split it with limits from -1 to 0 and 0 to ...
1
vote
0answers
30 views

Integral with Student's t density

Has anybody seen an integral of the following form or has some intuition on solving the following integral $$\int\limits_{ - \infty }^{q} {{t_\nu }\left( x \right)\log \left( {\frac{{{x^2}}}{\nu } + ...
3
votes
3answers
39 views

Improper integrals with u-substitution

I am going over improper integrals and I have a question about approaching a certain problem. The question is the the integral $$\int_1^\infty \frac{\log x}{x}\,dx$$ This integral does not converge, ...
3
votes
2answers
71 views

Improper integral of $\frac{\sin x}{x}e^{-ax}$

For $a>0$ define $$I(a)=\int_0^\infty \frac{\sin x}{x}e^{-ax} \, dx,$$ I can show it is continuous at $0$, but by differentiating in $a$, I can't see why $$I(a)=\frac{\pi}{2}-\arctan(a).$$ Thanks ...
2
votes
1answer
35 views

Improper Integral $\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx $

The question is : Does the following improper integral converges? $$\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx $$ I have tried some approaches but I'm not sure whether it was correct or not. ...