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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2
votes
2answers
56 views

Convergence of $\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$

Test the convergence of $$\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$$ Attempt: For sufficiently large $x$, we have $e^{-\sqrt x} > e ^{- x}$. I also tried solving the integral by By Parts ...
3
votes
1answer
30 views

Computing with Lebesgue integrals

This problem comes from Royden's Real Analysis, 4th ed., pg 84, #19: For a number $\alpha$, define $f(x)=x^\alpha$ for $0<x\le 1$ and $f(0)=0$. Compute $\int_0^1 f$. MY WORK: I know ...
1
vote
3answers
74 views

Convergence of $\int^\infty_0 \frac{e^{-\sqrt x}}{1+x}dx $

I would like to know if the improper integral $$\int^\infty_0 \frac{e^{-\sqrt x}}{1+x}dx \qquad (1)$$ is convergent or not. I tried substitution and integration by parts but got no simplification. So, ...
0
votes
1answer
51 views

Convergence of an integral $\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$

For what $\alpha\in\Bbb{R}$ does $\displaystyle\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$ converge ? The $0$ bound doesn't seem to be much of a problem, but I don't see how to deal with the ...
3
votes
2answers
73 views

Why does not $\int_{-\infty}^\infty x\,\mathrm{d}x$ converge?

It seems natural that it should converge, because for any $A\in\mathbb{R}$, $$\int_{-A}^A x\,\mathrm{d}x=\int_{-A}^0 x\,\mathrm{d}x+\int_0^A ...
2
votes
1answer
72 views

Improper integral: $\,\frac{1}{\pi}\int^\infty_0 \frac{\sqrt{x}}{1+x}e^{-xt}\,dx$

Good evening! How could one evaluate the following integral $$\frac{1}{\pi}\int^\infty_0 \frac{\sqrt{x}}{1+x}e^{-xt}\,dx$$ I have tried the substitution $x\equiv x^2$ but still I could not manage to ...
1
vote
1answer
71 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
0
votes
1answer
19 views

Compute an integral explicit (or explicit upperbound? Possible?)

I would just like to know if there is an explicit formula/computation for this integral: $$\int_{a}^1 \left(|x-a|^{\alpha} - |x-b|^{\alpha} \right)^2 dx$$ where $\alpha\in (-1/2,0)$ so the integral ...
2
votes
2answers
53 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$? According to Serge Lang, the integral on the left is the error term for Stirling's ...
3
votes
2answers
52 views

Asymptotic form of an integral

I would like to find an asymptotic form of the following integral when $s \to \infty$ ($s$ and $w$ are positive) \begin{equation} \int_{0}^{\infty} dx ~ \sqrt{x^2 + wx} ~ e^{-ixs} \end{equation} I ...
2
votes
1answer
33 views

Integral on $\mathbb{R}^d$

This is probably a simple question, but I don't have tons of experience integrating on $\mathbb{R}^d$ for arbitrary $d$. I'd like to compute the following integral $$ \int_{\mathbb{R}^d} ...
1
vote
2answers
35 views

Studying the divergence or otherwise of an improper integral

I'm supposed to study the following improper integral : $$\int^{1/2}_{-\infty}\frac{1}{1-x^{1/3}} dx$$ This could be an exercise out of 5 in our final exam paper, so I reckon there's a fast way to do ...
0
votes
1answer
41 views

what is being asked here?

I fail to see how this can be achieved: Define the improper integral (of a non-blocked function) as a limit, and calculate or prove that the integral diverges; $\large \int_0^1 \frac{dx}{ ...
3
votes
2answers
70 views

The value of $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$

Using the fact $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos nx dx=0$ ,find the value of $$\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$$ I tried through integrating by parts , ...
2
votes
3answers
136 views

Help solving an improper integral

I need to solve an improper integral which is: $$ \int_1^\infty \frac{2}{4{x^2}-1}dx $$ i was trying to solve it using simple substitution but cannot seem to figure it out, i tried a website to ...
3
votes
4answers
65 views

$\int_0^{\infty}y^2e^{-y} dy$

To calculate $\displaystyle \int_0^{\infty}y^2e^{-y} dy$ =$\displaystyle -y^2e^{-y}-2ye^{-y}-2e^{-y}|_o^{\infty}$ This should fail at $\infty$, but the answer is given as 2. Seems like $e^{-y}$ ...
4
votes
1answer
63 views

Does the integral $\int_0^\infty \sin(2x^4) \, dx$ converge absolutely/conditionally?

Does the integral $$\int_0^\infty \sin(2x^4) \, dx$$ converge absolutely/conditionally? I tried to evaluate $\int_0^b \sin(2x^4) dx$ by integrating by parts twice and got something relatively ...
2
votes
1answer
248 views

Improper integral (is it convergent?) (v 2.0)

Earlier today I asked about this question: Improper integral (is it convergent?) where the integral fortunately seems to be convergent. So we have that given $\alpha\in (-1/2,0)$ there is a $\gamma ...
2
votes
2answers
62 views

Improper integral (is it convergent?)

I would like to either prove or disapprove the following: Let $\alpha\in (-1/2,0)$ be given. Then we can find $\gamma \in (1,2)$ such that $$\int_0^1 \int_0^{u} ...
2
votes
4answers
257 views

Value of an unbounded definite integral

Evaluate the integration : $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(2x^{2}+2xy+2y^{2})}dxdy$$ The function is even about $x$ & $y$. So we can write, ...
0
votes
0answers
77 views

Derivative of a double integral (applying Leibniz rule)

I would like to differentiate the following expected value function with respect to parameter $\beta$: $$F(\xi_1,\xi_2) ...
2
votes
1answer
80 views

Proving that a function is Hölder-continuous

Let $\alpha\in(-1/2,0)$ and $x\in (0,1)$. Define the function $$f(x) = \int_x^1 z^{\alpha-\frac{1}{2}} (z-x)^{\alpha}dz.$$ I have the feeling that $|f(x)-f(y)| \leq C|x-y|^{1/2}$ or any other power ...
0
votes
0answers
23 views

Prove the convergence of an (improper) integral

I have proved the following: Let $t\geq 0$ and $\alpha \in (-1/2, 1/2)$ given. Then there exists a $\gamma \in (1,2)$ such that $$ \int_0^t \int_{0}^{s'} \frac{((t-s)^{\alpha}- ...
1
vote
0answers
79 views

Are there elementary ways to proof $\int_0^\infty \frac{\sin{x}}{x} \mathrm{d}{x}=\frac{\pi}{2}$? [duplicate]

Is there an elementary way to proof that $$\int_0^\infty \frac{\sin{x}}{x} \mathrm{d}{x} = \frac{\pi}{2}$$ or equivalently $$\int_{-\infty}^{+\infty} \frac{\sin{x}}{x} \mathrm{d}{x} = \pi \;\;\; ...
2
votes
2answers
115 views

Integral $\int\frac{(\sin(x))^2}{x^2+1} dx$

I have no idea of variable changement to use or other to calculate this integral : $$ \int_0^{\infty}\frac{(\sin(x))^2}{x^2+1}\,dx $$ Wolfram alpha gives me the result but really no idea ... I ...
10
votes
3answers
231 views

How to prove $\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$?

How to prove the following result? $$\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$$ For my part no idea?
2
votes
2answers
224 views

Conditions for the integral to equal zero

Can it be proven that for the integral $$\int_0^{\infty} e^{-x} f(x) dx $$ to equal zero, the function f (domain and codomain $\mathbb{R}$) has to be necessarily bounded?
9
votes
3answers
186 views

Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$

I'm interested in a closed form for this simple looking integral: $$I=\int_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$$ Numerically, ...
7
votes
0answers
120 views

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

I was playing around with the function $\dfrac{1}{1+x^2}$, and knowing that the integral over $(0,\infty)$ was $\dfrac{\pi}{2}$, I was hoping to see if there was some neat pattern to determining the ...
5
votes
3answers
137 views

Prove $\int_{-\infty}^\infty \cos t^2 dt = \sqrt{\frac{\pi}{2}}$

So, I need to prove the identity $$\int_{-\infty}^\infty \cos t^2 dt = \int_{-\infty}^\infty \sin t^2 dt = \sqrt{\frac{\pi}{2}}$$ and as a hint I have the Gaussian integral $$\int_{-\infty}^\infty ...
0
votes
1answer
24 views

Derivative of integral with infinity as upper bound

What is the solution to the derivative of following integral? I know how to take derivatives of integrals but I never came across one with infinity in one of his bounds. $F(t) = \int^{\infty}_t ...
5
votes
2answers
133 views

Difficulties understanding a proof of $\int_0^{\infty} \frac{\sin(x)}{x} \, dx = \frac{\pi}{2}$

I got a homework and I've trying to do this problem about 2 days, but I "lost my fight". So I turn to you. I have to prove that $$\int _0^\infty \frac{\sin (x)}{x} \, dx = \frac{\pi}{2}.$$ I can't use ...
2
votes
1answer
32 views

Derivation of Transformation for basic theta function

Given that $\vartheta(x) = \sum_{n = -\infty}^\infty e^{-\pi n^2 x}$, I am trying to finish a derivation that $\vartheta(x) = \frac{1}{\sqrt{x}}\vartheta(1/x)$. I believe that I am very close. I ...
1
vote
0answers
44 views

Is the following integral finite? for some values of the exponent?

This is a sequel of the question posted some hours ago: Is this integral finite? (convergent) Let us consider $\mathbb{R}^2$ and only the region $R=\{(x,y)\in \mathbb{R}^2: \, |x|>1, |y|>1\}$. ...
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
87 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
54 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
112 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
56 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
24 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
202 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
60 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
265 views