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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7
votes
4answers
119 views

show $\int^\infty_0 e^{-sx} x^{-1} \sin{x} dx = \frac14 \log{(1+4s^{-2})}$ for $s>0$

This is problem 2.6.58 of Folland's Real Analysis book: show $\int^\infty_0 e^{-sx} x^{-1} \sin{x} dx = \frac14 \log{(1+4s^{-2})}$ for $s>0$ by integrating $e^{-sx} \sin{(2xy)}$ over x and y. I ...
2
votes
0answers
15 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
0
votes
0answers
34 views

How to do contour integration? [duplicate]

I've searched a lot throughout the web, but havent found anything yet, so I am posting my own question. I am very interested in complex analysis, hopefully someone can help me out here. Suppose we ...
2
votes
1answer
49 views

Determine how large the number a has to be?

This is what i've done so far: i converted to limit notation lim as t goes to infinity of integral from a to t of 1/t^2+1 dt lim as to goes to infinity [arctan(t)] from a to t (lim as t goes to ...
0
votes
1answer
40 views

Integral of $\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$.

Can some one help me with the integral $$\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$$ According to my exercise I should be able to get $0$. Please help me .
10
votes
3answers
325 views

Closed form of $\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$

I have homework to evaluate this integral $$I=\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$$ Here is what I have done so far. I tried integration by parts using $u=\tanh(x)\,\tanh(2x)$ and ...
1
vote
3answers
71 views

Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge?

The Question Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge? Please note I have no knowledge of Alternating Series, Ratio and Root tests, Power Series, or Taylor and McLaurin Series. My Work ...
6
votes
3answers
109 views

How to evaluate this improper integral?

I got stuck when evaluating these two improper integrals:$$ \int_a^b\frac{dx}{\sqrt{(b-x)(x-a)}} $$ and$$ \int_0^1\frac{dx}{\sqrt{x-x^3}} $$ How to evaluate them? Thank you!
30
votes
2answers
545 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
1
vote
1answer
29 views

How do you evaluate an exponential term that contains both $-\infty$ and $+\infty$?

What does $\int_{0}^{\infty} e^{y(iu-\alpha)}dy = ?$ Please note $i$ is a complex variable, $\alpha$ and $u $ are constants. I know this integral evaluates to: ...
3
votes
3answers
148 views

How to calculate $\int_0^\infty \frac{dx}{1+x^6}$ [duplicate]

Whenever I tried to do, it failed. Is there anyone to help? $$\int_0^\infty \frac{dx}{1+x^6}$$
0
votes
2answers
36 views

Why is the improper integral $\int^0_{-\infty}{1\over 3-4x}dx$ divergent?

Take the following: $$\int^0_{-\infty}{1\over 3-4x}dx$$ Substituting $t$ for $-\infty$, we can replace the above with $$\left.\lim_{t\to-\infty}-{1\over4}\ln(3-4x)\right\rbrack^0_{t}\ ,$$ which ...
0
votes
0answers
11 views

Uniform convergence and Riemann integration (including improper integrals)

Let, for each $n\in\mathbb{N}$: $f_n:S\to \mathbb{R}$ a Riemann-integrable function over $S$ where $S\subseteq \mathbb{R}^N$ is bounded. Suppose that $f_n$ converges uniformly to a function $f:S\to ...
3
votes
2answers
89 views

Evaluating a double integral from zero to infinity

How do I evaluate this integral? I don't understand at which point the limit notation should set in? And my method yields $0$ in the end. The integral is: $$ \int_0^{\infty} \int_0^{\infty} ...
3
votes
1answer
50 views

Real integral giving a complex result

I have to solve the following integral $$ \int_{-\infty}^{\infty} dx \frac{e^x-1}{x} e^{-\frac{(x+y)^2}{2 \sigma^2}}$$ This integral converges, becaue $\frac{e^x-1}{x}$ goes to 1 for $x \to 0$. ...
7
votes
2answers
150 views

How to find closed-form of $\int_{0}^{+\infty} \operatorname{sech}^2 (x^2)\,dx$

How to find this integral closed form: $$I=\int_{0}^{+\infty}\operatorname{sech}^2{(x^2)}\,dx$$ where $\operatorname{sech}{(x)}$ is defined as secant of hyperbolic function. This problem ...
1
vote
2answers
41 views

Integral with conditions

Compute $$\displaystyle\min_{a,b,c} \displaystyle\int_{0}^{\infty} \left | x^3-a-bx-cx^2 \right |^2e^{-x}\, dx$$ Please, any suggestions are welcome. Thanks you all.
1
vote
3answers
44 views

Positive to negative infinity integration

$$\int_{-\infty}^{\infty} xe^{-x^2} dx$$ I guess let $u = -x^2$ , hence $\frac{du}{dx}=-2x$ $$dx = \frac{-1}{2x}du$$ $$\int_{-\infty}^{\infty} \frac{e^{u}}{2} du$$ this is where i get lost
2
votes
1answer
127 views

Applying the Mean Value Theorem to improper Riemann integral(homework)

The definition for the Mean Value Theorem I am working with: If a function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, Then there exists at least one point $c$, where $a<c<b$ such ...
0
votes
1answer
40 views

Convergence of $\int_1^{+\infty}\frac{\sin(x)}{x}\arctan(x) \mathop{d}x$

How to prove that integral $$\int_1^{+\infty}\frac{\sin(x)}{x}\arctan(x) \mathop{d}x$$ converges (or does not)?
1
vote
1answer
33 views

Prove that $\int_0^\infty y^{\frac12} e^{-y^2} \int_0^\infty y^{-\frac12} e^{-y^2} =\frac{\pi}{2^{\frac32}}$

My attempt: $\int_0^\infty y^{\frac12} e^{-y^2}$ $=\frac12 e^{-z} z^ {-\frac14} dz$, using the transformation: $y^2=z$, i.e. $y=z^\frac12$ $=\frac12 e^{-z} z^ {1-\frac54} dz$ $=\Gamma\frac54$ ...
1
vote
1answer
71 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
0
votes
1answer
30 views

Calculate this integral in $N$-dimensional space

I want to calculate the integral $$\int_{\mathbb{R}^N \times \mathbb{R}^N} \chi_{[0,E]}\left(\sum_{i=1}^N \frac{p_i^2}{2m} + \frac{m \omega^2 q_i^2}{2} \right) \,dp\, dq.$$ Now I should explain what ...
7
votes
3answers
132 views

Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$

In my course, I have to prove formula below $$I=\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$$ for $a,b,c>0.$ I know that ...
3
votes
2answers
113 views

converting improper double integrals to polar form: what do I do with infinity limits

I need to convert $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}-e^{\frac{x^2+y^2}{5}}dA$$ To polar form. I know $x^2+y^2 = r^2, $ and $dA = rdrd\theta$ But what do I do with the $\infty$ ...
11
votes
4answers
267 views

Finding $\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$

How do we prove that $$I(m)=\int_{0}^{\pi/2} \frac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x=\frac{\log{m}}{m^2-1}$$ I see that $$I(m)=\frac{\partial}{\partial m} \int_{0}^{\pi/2} \arctan({m\tan x}) \ ...
1
vote
1answer
40 views

Proof about Simple random walk in $\mathbb{R}^{d}$

I have read something about random walks in $\mathbb{R}^{d}$. The random walks is assumed to be started at origin. There is a theorem said that For $d =1$ or $2$, the random walk is recurrent. (i.e. ...
3
votes
1answer
46 views

Show that $\int_0^r \frac{\mathrm{d}t}{\sqrt{r^2 - t^2}} $ is independent of $r$

I'm trying to show that the integral $$\int_0^r \frac{1}{\sqrt{r^2 - t^2}}\mathrm{d}t$$ is independent of $r$, without using trigonometric functions (namely, $t=\cos s$ and such). Can it be done? ...
0
votes
1answer
23 views

CDF, PDF, and integral with respect to a function

In a certain textbook, I see the Cumulative Distribution Function (CDF) of a continuous random variable X defined as $$\int_{-\infty}^{x'} dp(x)$$ where p(x) is the Probability Density Function of X. ...
10
votes
2answers
120 views

Finding $\displaystyle \int_{1}^{\infty} \frac{\sin^4(\log x)}{x^2 \log x} \mathrm{d}x$

How do we prove that $$\int_{1}^{\infty} \frac{\sin^4(\log x)}{x^2 \log x} \mathrm{d}x=\dfrac{\log\left(\dfrac{625}{17}\right)}{16}$$ I tried substitutions like $\log x=\arcsin t$, but it doesn't ...
0
votes
2answers
54 views

Show that $\int_0^{\infty}\dfrac{\ln x}{1+x^2}\hspace{1mm}dx=0$ [duplicate]

Show that $\int_0^{\infty}\dfrac{\ln x}{1+x^2}\hspace{1mm}dx=0$ The problem I am facing is with the hint. The hint says, use the substitution $u = 1/x$, it makes no sense to me. Why would we use ...
5
votes
3answers
80 views

Finding $\int_{1}^{\infty} \frac{x 3^x}{(3^x-1)^2}$

How do we prove that $$\int_{1}^{\infty} \frac{x \cdot 3^x}{(3^x-1)^2} \mathrm{d}x=\dfrac{3\log{3}-2\log{2}}{2\log^2{3}}$$ I have tried some substitutions such as $3^x=t$, but it didn't work out. The ...
5
votes
1answer
91 views

Show $\int_0^\infty f\left(x+\frac{1}{x}\right)\,\frac{\ln x}{x}\,dx=0$ if $f(x)$ is a bounded non-negative function

Lemma If $f(x)$ is a bounded non-negative function, then \begin{equation}\int_0^\infty f\left(x+\frac{1}{x}\right)\,\frac{\ln x}{x}\,dx=0\end{equation} I found this lemma on internet and ...
8
votes
3answers
257 views

Prove $\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}$

How does one prove the following integral \begin{equation} \int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4} \end{equation} Wolfram Alpha and Mathematica can ...
7
votes
1answer
77 views

Finding a double integral $\int_1^\infty\int_0^\infty\frac{1}{(x^3+y^3)^3}\mathrm{d}x\ \mathrm{d}y=\frac{10\pi}{189\sqrt3}$

How do we prove that $$\int_1^\infty\int_0^\infty\dfrac{1}{(x^3+y^3)^3}\mathrm{d}x\ \mathrm{d}y=\dfrac{10\pi}{189\sqrt3}$$ I tried to expand and use partial fraction, but in vain. I don't have a clue ...
1
vote
1answer
47 views

Am I doing something wrong with this improper integral?

I have a little discussion with my friends about my "resolution" and calculation of $$\int_{-\infty}^1 e^{4x} \, dx.$$ I did $$\int_{-\infty}^1 e^{4x} \, dx =\int_{-1}^{\infty} e^{-4x} \, dx = ...
14
votes
2answers
156 views

A Mathematical Coincidence, or more?

According to the paper "Ten Problems in Experimental Mathematics", $$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times ...
8
votes
2answers
129 views

Finding the closed form of $\int_0^1 \frac{(1-x+x\log x)\operatorname{Li}_3(x)}{x(x-1) \log x} \ dx$

$\def\Li{{\rm{Li}}}$Here I have a question I just received, and still trying to find a proper starting point $$\int_0^1 \frac{(1-x+x\log x)\Li_3(x)}{x(x-1) \log x} \ dx$$ What starting point would ...
2
votes
0answers
31 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...
7
votes
1answer
242 views

How to solve this complicated double integral problem?

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
11
votes
1answer
112 views

A limit evaluating to $2 K$ (Catalan's constant)

Experimentally I discovered the limit below that says that $$\lim_{n\to\infty} \int_0^{\pi/2} \frac{1}{\displaystyle ...
21
votes
2answers
309 views

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ...
5
votes
0answers
82 views

Finding the closed form of $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k n}$

A while ago I computed pretty easily the series $\sum_{k=1}^{\infty} \sum_{n=1}^{\infty}(-1)^{k+n} \frac{\log(k+n)}{k+ n}$ and then I thought of tackling the case where we have the product instead of ...
3
votes
0answers
65 views

A hard integral from probability theory

I am trying to resolve this integral, which comes out of considering a compound distribution of normal variables: $$ \int_{-\infty}^{\infty} \frac{1}{\sigma_{\sigma} \sqrt{2 \pi}} ...
8
votes
5answers
183 views

Improper integral of $\dfrac{x^2}{ e^x−1}$

I am trying to evaluate $$\int_0^t \frac{x^2}{ e^x−1}dx$$ for $t>0$. I tried integrating by parts, like follows: $$\int_0^t x\cdot\frac{xe^{-x}}{1-e^{-x}}dx=t\cdot Li_2(1-e^{-t})-\int_0^t ...
3
votes
1answer
89 views

Differentiation under integral

I want to compute the following integral, $\int_0^\infty\frac{dx}{(x^2 + p)^{n}}$ for any $n \in \mathbb{N}$ I've tried to add a parameter, obtaining its integral and then taking derivatives with ...
5
votes
1answer
60 views

Three integral involving polylogarithm function

$\newcommand{\Li}{\operatorname{Li}}$Evaluate the following integrals $$\int\limits_0^1 \frac{\Li_2^3(x)}{x}dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_3 (x)}{x} dx, \quad \int\limits_0^1 ...
0
votes
1answer
53 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
9
votes
2answers
214 views

Closed form of $\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$

What real analysis tools would you recommend me for getting the closed form of the integral below? $$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$
8
votes
3answers
234 views

Using differentiation under integral sign to calculate a definite integral

I want to calculate the integral $$\int^{\pi/2}_0\frac{\log(1+\sin\phi)}{\sin\phi}d\phi$$ using differentiation with respect to parameter in the integral ...