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
votes
1answer
53 views

Is this integral finite? (convergent)

I came to this problem and I got very curious to know... intuitively I would say this integral is not finite but maybe it is. Let us consider $\mathbb{R}^2$ and only the region $R=\{(x,y)\in ...
0
votes
2answers
51 views

How to compute $I_n=\int^{+\infty}_{0}x^ne^{-x}dx,\ n \geq 0$? [closed]

I want to compute the following integral: $$I_n=\int^{+\infty}_{0}x^ne^{-x}dx,\ n \geq 0$$ How to do it?
3
votes
0answers
88 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
1
vote
2answers
59 views

Definite integration involving square root function

How to integrate this definite integral: $$\int_{0}^{\pi/2} \big(\sqrt{\cos x}+ \sqrt{\cot x}\,\big)\,\mathrm dx$$
2
votes
3answers
113 views

Convergence of $\int_2^\infty \frac{dx}{x^2 \cdot (ln(x))^{\alpha}}$?

For which values of $\alpha > 0$ is the following improper integral convergent ?: $$\int_{2}^{\infty}{{\rm d}x \over x^{2}\ \ln^{\alpha}\left(\, x\,\right)}$$ I tried to solve this problem by ...
6
votes
2answers
135 views

Computing a nasty integral (probably with computer algebra system)

I'm trying to do this integral, not sure if it is possible: $$ \int_{1}^{\infty}\int_{0}^{\infty} \exp\left(\, -\,{x^{2} \over y^{2}}\,\right) \exp\left(\,-\,{y^{2} \over z^{2}}\,\right) \exp\left(\, ...
1
vote
1answer
29 views

Limit of improper integral

I have a function $F(x) = \frac{1}{x} \int_0^x f(t) \, dt$. If $f(x) \to L$ show that $F(x) \to L$ as $x\to\infty$. So far i have tried to split the integral up like so $$ ...
1
vote
2answers
57 views

Orientation of multiplying integrals

Consider, $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ The trick is to multiply by $I$ again to get $I^2$ But they often write: $$I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2 - ...
1
vote
1answer
25 views

Integral $\int_0^\infty |x-c|e^{-2x}dx$

I have to evaluate the integral: $$\int_0^\infty |x-c|e^{-2x}dx$$ with c $\in \mathbb{R}$. I would evaluate the integral this way: http://math.ucr.edu/~jmd/9B_S14_AbsInt.pdf. This would give me one ...
1
vote
2answers
66 views

An improper integrals related to probability, $\int_0^\infty\frac1y \exp(\frac{-x_0}y-y)\,dy$

How can I calculate the integral $$\int_0^\infty{\frac1y e^{\frac{-x_0}y-y}}dy$$ in terms of well-known constants and functions? I used some fundamental techniques of integration but got nothing.
4
votes
6answers
215 views

Prove that $\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}$

Good evening everyone, how can I prove that $$\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}\;?$$ Well, I know that $\displaystyle\frac{1}{x^4+x^2+1} $ is an even function ...
2
votes
2answers
69 views

Improper integral comparison test $\frac{\sin(x)}{x^2}$

The question asks whether the following converges or diverges. $$ \int_{0}^{\infty } {\left\vert\,\sin\left(\,x\,\right)\,\right\vert \over x^2}\,{\rm d}x $$ Now I think there might be a trick with ...
0
votes
2answers
58 views

Convergence of $\int_{1}^{\infty} \frac{\sin x}{x^{\alpha}}dx$

For which values of $\alpha > 0$ is the following improper integral convergent? $$\int_{1}^{\infty} \frac{\sin x}{x^{\alpha}}dx$$ I tried to solve this problem by parts method but I am nowhere ...
2
votes
2answers
112 views

How to find $\int_{2}^{\infty} \frac{dx}{x^2 \ln^\alpha(x)}$

What would be the integration of the given function: $$\int_{2}^{\infty} \frac{dx}{x^2 \ln^\alpha(x)}$$ I have tried to solve this question by substitution and by parts also but non of them seems ...
4
votes
3answers
86 views

Convergence of $\int_0^\infty $sin$ (x^p) dx$

Consider the $\displaystyle \int_0^\infty $sin$ (x^p) dx$. Does it converge when $p<0$? Does it converge when $p>1$? My Work: Let $x^p=y$, then $\displaystyle \int_0^\infty $sin$ ...
5
votes
2answers
267 views
1
vote
2answers
86 views

Convergence of $ \int_{-\infty}^\infty \cos(x\log(\lvert x\lvert ))\,dx $

Show that the improper integral $$ \int_{-\infty}^\infty \cos(x\log(\lvert x\lvert ))\,dx $$ is convergent. I rewrote it, using even function symmetry of cosine, as twice the integral from zero to ...
0
votes
1answer
34 views

How do I test the convergence/divergence of the following improper integral?

$$\int_0^1 \frac{dx}{(x-1)^{2/3}}$$ I also know of the result: if $lim_{x \to b} (b- x)^r f(x)= A \not= \infty$, then $\int^b_af(x) dx$ converges. But this requires I take $-1$ out of the brackets, ...
1
vote
1answer
32 views

Definite integral of a function containing max()

I want to determine the constant c so that this definite integral evaluates to 1. I have no information about the behaviour of this function in different intervals (e.g. which term inside max() is ...
2
votes
1answer
107 views

Evaluating$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $using residues

I need help to solve the next improper integral using complex analysis: $$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $$ I have problems when I try to find residues for the function $ f = ...
5
votes
4answers
116 views

Show that all derivatives of $f(x) = \frac{\sin(2\pi x)}{\pi x}$ belong to $L^2(\mathbb{R})$

I would like to show that all derivatives of $f(x) = \frac{\sin(2\pi x)}{\pi x}$ belong to $L^2(\mathbb{R}).$ One route I have tried is a power series approach. If we expand our function $f$ in a ...
3
votes
1answer
48 views

Cauchy P.V. Of an improper inegral

the poles are $x=+1,-1,i,-i$ we should take only the upper have of axis so we should take residue of $1$ and $i$? right in this problem the book took only $x= i$. I don't know why !! please help
0
votes
1answer
24 views

Improper integral with two infinite bounds

How can I solve: $\int^\infty_{-\infty} \frac{1}{y^2+1} dy$ I have tried splitting it up in two limits: $\lim\limits_{n \to \infty} [arctan(y)]^n_0 + \lim\limits_{n \to \infty} [arctan(y)]^0_n$ ...
6
votes
2answers
99 views

Evaluating $\int_0^\infty \sqrt{\frac{x}{e^x-1}}dx$ in terms of special functions

Introduction: I've been studying integrals of the form $$\int_0^\infty \frac{x^a}{(e^x-1)^b}dx$$ where a and b are real parameters. I've been able to find closed forms for the integral in terms of the ...
4
votes
1answer
170 views

How to calculate the two integrals?

How calculate the following integrals $$I=\int_0^{\pi/2}x^2\frac{(1-\tan\,x)\sin4x}{\sqrt{\tan x}}dx$$ $$K=\int_0^{\pi/2}x^2\frac{(1+\tan\,x)\sin4x}{\sqrt{\tan x}}dx$$ EDIT It seems that one ...
0
votes
0answers
25 views

Determining Normalization Constant for a PDF when the integral to be evaluated is a divergent integral

Suppose, we assign the probability to a vector as inversely proportional to the maximum of its values across all dimensions (e.g. max(x1,x2,x2) if its a 3-D vector). If we want to give a mathematical ...
0
votes
1answer
31 views

Finding the residue of the improper integral $\frac{1}{z^4+4}$

$$f(z) = \frac{1}{z^4+4}$$ the roots of this are: $z^2=\pm i\sqrt{2} \implies z=\pm\sqrt{i\sqrt{2}}$ and $z=\pm i\sqrt{i\sqrt{2}}$ i.e. $$f(z) = \frac{1}{(z\pm\sqrt{i\sqrt{2}})(\pm ...
2
votes
3answers
109 views

If $\lim_{x\to\infty}\int_0^x a(t)\,dt=L$, is it true that $\lim_{x\to\infty}a(x)=0?$

Let $s(x)=\int_0^xa(t)\,dt$. I wonder: If $\displaystyle \lim_{x\to\infty}s(x)=L$ is it true that $\displaystyle\lim_{x\to\infty}a(x)=0?$
9
votes
3answers
292 views

What are other methods to evaluate $\int_0^1 \sqrt{-\ln x} \ \mathrm dx$

$$\int_0^1 \sqrt{-\ln x} dx$$ I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral. $$y=-\ln x$$ $$\bbox[8pt, border:1pt solid ...
0
votes
0answers
43 views

Contour integration when pole is outside the contour

Here they are using the pole OUTSIDE the contour? I thought this was illegal according to the residue theorem or we are not supposed to do contour integration with poles outside the contour itself.
2
votes
1answer
48 views

How many poles have to be inside the contour?

If we consider $$\int_{0}^{\infty} \frac{dx}{1+x^2}$$ Using complex contour integration only. We choose a contour in the TOP HALF plane. From the poles $z = \pm i$ only, the pole: $z=i$ is ...
1
vote
1answer
26 views

Limits for each part of the improper integral, why?

My handbooks "says" that a limit must be calculated for each part of the improper integral. When using the same limit for all parts, it is called the Cauchy Principal Value. My question is, what the ...
3
votes
2answers
81 views

How do I evaluate the following integral $\int_{-\infty}^{\infty} e^{-\sigma^2 x^2/2}\; \mathrm dx$? [duplicate]

How do I evaluate the following integral $$\int_{-\infty}^{\infty} \exp\left(-\frac{\sigma^2 x^2}{2}\right) \mathrm dx\;?$$ How is it even possible to find an antiderivative? The integral is ...
2
votes
1answer
58 views

Solution to the exponential integral $\int_{0}^{\infty}\exp\bigl(-pR-\frac{ER^{-f}}{B}\bigr)dR$

I have a exponential integral $$\int\limits_{0}^{\infty}\exp\left(-pR-\frac{ER^{-f}}{B}\right)dR$$ where E, p, B, and f are constants, $E>0$ Context This equation arrives when I try to consider ...
5
votes
1answer
67 views

Evaluation of $\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d\!}x$ for $\mu>0$

I'm trying to evaluate the integral $$ \Psi(\mu,\nu)=\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d}\!x\qquad(\text{for}\; \mu>0)\tag 1 $$ where $\nu\in\Bbb R$ and $f$ ...
0
votes
1answer
69 views

How to check if functions are integrable?

Consider two functions $$ \int_0^1 \frac{1}{e^x-1} dx $$ and $$ \int_0^1 \frac{1}{(e^x-1)^2} dx $$ How to check if these functions are integrable?
0
votes
1answer
37 views

Discussing the convergence of $\displaystyle\int_I\frac{x+2}{\sqrt x\left(x^2+x+1\right)^4}\mathrm dx$

Let $$f(x) = \frac{x+2}{\sqrt{x}\left(x^2 + x + 1\right)^4}$$ Discuss the convergence of $\displaystyle\int_0^1f(x)\,\mathrm dx$ and $\displaystyle\int_1^{+\infty}f(x)\,\mathrm dx$. I encountered ...
1
vote
4answers
105 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
4
votes
2answers
189 views

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) - \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} ...
1
vote
1answer
90 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
5
votes
4answers
261 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
7
votes
2answers
143 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
2
votes
1answer
38 views

Why $ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $?

from this answer I could not see what is happening here: $$ \int_0^{\infty} du \, \frac{e^{-3 u} - e^{-4 u}}{u} = \int_0^{\infty} du \, \int_3^4 dt \, e^{-u t} \\ $$ What technique of integration ...
6
votes
1answer
187 views

How evaluate the following hard integrals?

Prove: $$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{\cos2x}{2\sin^2x}}}dx=\frac{\pi}{96}[{\pi^2}-6\ln^22]$$ And ...
3
votes
6answers
162 views

Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions

Let $$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$ How to show that it converges with no use of trigonometric functions? (trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be ...
1
vote
0answers
33 views

$\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$

I was thinking about $\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$ The inspiration came from the following 3 integrals : Lemma If $f(x)$ is a bounded non-negative ...
0
votes
1answer
30 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
2
votes
1answer
86 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
12
votes
4answers
239 views

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

Nowadays I encounter an integral which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} ...