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
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1answer
65 views

For which $a>0$ does $\int_a^\infty \frac{\mathrm{d}x}{(x^2-a)^{4a}}$ converge?

As the title suggests, I need help finding $a>0$ for which the following improper integral converges: $$\int_a^\infty \frac{\mathrm{d}x}{(x^2-a)^{4a}}$$ So, at first I thought I would just do ...
0
votes
1answer
17 views

Contour integral to real integral: find suitable change of variables

There's probably simple solution but... I have a contour integral of the form $\int _{-i \infty}^{+i \infty} f(t) \ dt$. I want to make a transformation $t = g(s)$ so that the integral is real and of ...
0
votes
1answer
43 views

Value of integral

The value of $$\int_0^\infty t^{-3/2}\left(1-e^{-t}\right)\,dt=$$ Source. Applying integration by parts or using any kind of substitution is not working. My attempt: Splitting the ...
2
votes
2answers
61 views

Integral of Lorentzian type with trigonometric function

Consider the following Riemann integral $$ \int_0^\infty \mathrm{d}x \frac{\alpha^2}{(x-x_0)^2+\alpha^2} \frac{\sin\left[{\left(x - x_1\right) t }\right]}{x-x_1} $$ with the displacements $x_0,x_1 \in ...
1
vote
2answers
55 views

Comparing the Indefinite Integrals Convergence for $1/x$ and $1/x^{2}$ between 1 and $\infty$.

This is my question. I've been told that $1/x^2$ converges while $1/x$ diverges. My intuition tells me that looking at these just plain out as functions that both should converge...my reasoning is as ...
5
votes
3answers
79 views

Is the integral $\int e^{2 \pi i z^2} dz$ uniformly bounded for any interval of $\mathbb{R}$?

I was wondering if there exists a constant $C$ such that $| \int_I e^{2 \pi i z^2} dz | \leq C $ for any interval $I$ of $\mathbb{R}$? Here I want $C$ to be independent of the choice of the interval ...
0
votes
0answers
40 views

Bound on the following integral using integration by parts

Let $W: \mathbb{R}^s \rightarrow [0,1]$ be a smooth function supported on $[0,1]^s$ that satisfies $$ \left| \frac{\partial^k }{\partial x_{i_1} \cdots \partial x_{i_k}} W(\mathbf{x}) \right| \leq ...
1
vote
1answer
43 views

Proof that $ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3)$, for $2<\Re(d)<4 $?

Can one prove that $$ \int_0^\infty x^{d-4}\sin x\, dx = \cos \frac{\pi d}{2} \Gamma(d-3),\text{ for }2<\Re(d)<4? $$ I would prefer using the methods of contour integration.
1
vote
1answer
28 views

How to prove that: $\int_{0}^{\infty} \frac {1}{\frac 1 2(e^x-e^{-x} )\cdot x} dx$ Diverges.

How to prove that: $\int_{0}^{\infty} $$ \frac{1}{\sinh(x)\cdot x }dx=\int_{0}^{\infty} \frac {1}{\frac 1 2(e^x-e^{-x})\cdot x} dx$ Diverges. SOLUTION ATTEMPT: I thought about separating this ...
3
votes
1answer
102 views

Uniformly boundedness of an oscillatory integral

Let $f\in H^1(\mathbb{R}^3)$. Define, for $M>0$, $$I(M)=\int_{B(0,M)}e^{i|y|^2}f(y)dy$$ where $B(0,M)$ is the ball centered in the origin and of radius $M$ in $\mathbb{R}^3$. Is it true that ...
0
votes
0answers
20 views

A double integral with parameters

Please help me solve this integral: $$\frac{1}{\pi}\int\int_{-tz \in \left[ 0,\frac{\pi}{2} \right]} \cos(tz)\left[ (x-z)\left( e^{-t^{\alpha_1}}-e^{-t^{\alpha_2}} \right) \right] \, dt \, dz$$ ...
0
votes
1answer
45 views

Improper integral of $\frac{1}{x} \sin \frac{1}{x}$

Let $f(x)= \frac{1}{x} \sin \frac{1}{x}$ when $x \in (0, 1]$and $f(0)=0$. Prove that $\int_{0}^1 f(x)dx$ exists. Can someone give me a hint to solution?
3
votes
1answer
54 views

Prove that $\int_{2}^{\infty} \frac{dx}{x^{1-\epsilon}\cdot \ln(x)^{\beta}}$ diverges for every $\beta$.

let $\beta,\epsilon\in \mathbb R$, such that $\epsilon>0$. prove that for every $ \beta$: $$\int_{2}^{\infty} \frac{dx}{x^{1-\epsilon}\cdot \ln(x)^{\beta}}$$ Diverges. SOLUTION ATTEMPT: if ...
3
votes
1answer
48 views

Elliptical Integral that diverges at one point

I have to solve the following integral $$I=\int_{\lambda_1}^yd\lambda\frac{1}{1-\lambda}\sqrt{\frac{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_4)}{\lambda-\lambda_3}}$$ where ...
2
votes
1answer
70 views

Closed form of $\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx, \end{equation} where $A$, $B$, and $C$ are ...
3
votes
2answers
85 views

Is there a simple proof for $\int_1^{\infty}\frac{2x^2\log^2 x}{(x^2-1)^2}dx=\frac{1}{4}(7\zeta(3)+\pi^2)$?

This morning I've computed easy computations with simple integral representations for Apéry constant and I find a (conjecture) formula using an online integrator (Wolfram Alpha), I woluld like if it ...
2
votes
0answers
48 views

Improper definite integral $\int_0^\infty e^{-x^2}(x + k)^\alpha dx$

I am unsure how to calculate the following definite integral: $$ \int_0^\infty e^{-x^2}(x + k)^{\alpha}dx,$$ where $k > 0$ and $\alpha$ is a real number. I tried integrating by parts and also a few ...
2
votes
2answers
46 views

Improper integral involving trigonometric function

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$. The example I have in mind is ...
1
vote
2answers
68 views

Upper bound of $\int_{-\infty}^{\infty}\sin(x)dx$. [duplicate]

From another question, improprer integral $$\int_{-\infty}^{\infty}\sin(x)dx$$ is not $$\lim_{a \to \infty} \int_{-a}^a \sin x \, d x$$ but $$\lim_{a \to \infty}\lim_{b \to \infty}\int_{-a}^b \sin x ...
0
votes
0answers
19 views

Improper integrals $\int_{-\infty}^{+\infty} \cos t dt$ and $\int_{-\infty}^{+\infty} \sin t dt$ [duplicate]

What can we say about following improper integrals? $$\int_{-\infty}^{+\infty} \cos t dt, \ \ \ \ \ \ \int_{-\infty}^{+\infty} \sin t dt$$ My attempt: $$\int_{-\infty}^{+\infty} \cos t dt=2 ...
0
votes
3answers
89 views

How does the integral $\int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x$ converge?

I tried using the fact that $\ln(f(x)) < f(x)$ but that doesn't seem to work. It's an improper integral. $$ \int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x $$
2
votes
4answers
114 views

How to show $\int_{1}^{\infty} \frac{\sin^2(x)}{x^2}dx$ is finite?

At first, my approach was to directly take the improper integral of it. However, it seems not that easy. Then I tried to find another fraction to make a comparison. I got $\frac{\sin^2(x)}{x^2} ...
1
vote
1answer
56 views

Related integral problem to the Gaussian integral

So according to Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$, $$\int_0^\infty e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$ I want to solve for this. $$\int_0^\infty e^{-x^2}\ln(x)dx$$ ...
2
votes
2answers
88 views

How do I evaluate $\int_{0}^{\infty} u^{z-1}(e^{iu}-1) \, du$?

I am trying to evaluate the following integral that shows up in this paper http://arxiv.org/pdf/1103.4306v1.pdf $I=\int_{0}^{\infty} u^{z-1}(e^{iu}-1)du= \Gamma(z)e^{\frac{iz\pi}{2}}$ for ...
2
votes
0answers
52 views

$1 = \int f(x) \ dx$, by definition, or by Lebesgue's theorem?

We have that (in the context of Lebesgue integration)$$\lim_{n \rightarrow \infty} \int_{-\infty}^n f(x) \ dx = 1$$ I wish to show that this implies $\int_{-\infty}^\infty f(x) \ dx = 1$. Is this true ...
0
votes
0answers
24 views

A multiple of the Gamma Function when integrated between $0$ and $\infty$

I was reading through this answer on stats.stackexchange, but didn't follow the mathematics behind one step. They have $$\int\limits_{\tau=0}^{\infty} e^{-\tau( ...
0
votes
0answers
28 views

Maximizing an Integral Quantity

Consider the function $f(x) = x^{-\ln x}$. Let $U(x)$ be a function such that $U'(x) > 0$ and $U(x) < x$. Suppose that $\int_0^\infty U(f(x)) = T$. What function $U$ maximizes the quantity ...
0
votes
1answer
28 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
0
votes
1answer
30 views

Test for convergence of integral

Considering the behaviour of the integrant at both integration limits, study the convergence of the integral: $$\int_{0}^\infty x \sin\left(\frac{1}{x^\frac{3}{2}}\right). $$ I was trying to ...
0
votes
1answer
30 views

Same integral diverges for different limits?

I was investigating convergence of such an integral: $\int_{1}^\infty $$\frac{dx}{x(1+x)} $ I used comparison test: $\int_{1}^\infty $$\frac{dx}{x(1+x)} $ < $\int_{1}^\infty $$\frac{dx}{x^2} $ ...
2
votes
0answers
30 views

Integral of $\sin (e^{x^2})e^{-x^2+ix\lambda}$

Trying to solve this problem : Is $T$ invariant under Fourier transform ? Where : $T= \{f\in \mathcal{C}^{\infty} (\mathbb{R}), \forall n \in \mathbb{N}, |x|^nf(x) \to 0 \; \text{when}\; |x| \to ...
0
votes
1answer
38 views

Convergence and estimate of improper integral involving function $\frac{\sin\pi x}{\pi x}$.

Study the convergence of the improper integral $$\int_{-\infty}^\infty \frac{d^k}{dx^k}\left\{\frac{\sin(\pi(x-t))}{\pi(x-t)}\right\}\Biggl|_{x=n}\ \cdot \ ...
2
votes
2answers
96 views

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform?

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform? I had an idea in my mind. To use the $\text{sinc}$ function and take its inverse Fourier Transform or something like that. ...
0
votes
1answer
41 views

Determine if the improper integral converges

I'm having trouble proving that this improper integral converges, if it does. $$\int_3^{\infty} \frac{dx}{x+e^x}$$
0
votes
1answer
72 views

When is $\int_1^\infty \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\,\mathrm{d}x$ finite?

How do I solve the following problem? For which of the following values of $a$ is the integral $$ \int_1^\infty \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\,\mathrm dx $$ finite? $$ a ...
0
votes
5answers
109 views

Evaluating the integral $\int_{-\infty}^\infty x^2\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \ dx$

How does one evaluate $$\int_{-\infty}^\infty x^2\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \ dx ?$$ The result is $1$ and it corresponds to $E[X^2]$, where $X$ is a random variable with ...
1
vote
2answers
48 views

How do I change integration order?

I need to bound (or calculate in a closed form) this integral: $$\int_{R}^{\infty}dz\int_{a-bz}^{\infty}dy\cdot \frac{1}{\sqrt{2\pi}}e^{\frac{-y^2}{2}} e^{-z}$$ as a function of $R,a,b$. The result ...
0
votes
0answers
32 views

Integrating this (rather messy) integral and determining the limiting behaviour

From a preceding question I have worked on, an integral to evaluate: $$\frac{2i\sqrt{a}}{(a^2+1)}\cdot\int_{-\infty}^\infty k\cdot\exp\bigg[i\bigg(k-\frac{k^{2}ht}{2m}\bigg)\bigg]~dk$$ I need to ...
1
vote
1answer
34 views

Integrating an integrand with an absolute value on exponential

This is one heck of an embarrassment but it is amazing how these bits of subtlety gets lost in the back of the head after the first year of undergraduate studies-with every computation chucked into ...
3
votes
2answers
70 views

Is the improper integral $\int_0^{\pi/2} \sqrt{\cot x}\, dx$ convergent?

Is the improper integral $\int_0^{\pi/2} \sqrt{\cot x} \,dx$ convergent? I am unable to use any kind of comparison test or anything.
4
votes
1answer
123 views

Evaluating the improper integral $\int_0^{\infty}\frac{dx}{1+x^3}$

Evaluate $$\int_{0}^{\infty}\frac{dx}{1+x^3}.$$ I tried integration by partial fraction. My work is below: $$\int_{0}^{\infty}\frac{dx}{1+x^3}=\frac{1}{3} ...
3
votes
1answer
49 views

Convergence of an integral $\int_1^{+\infty} \frac{\ln^2(1+x)}{x^{2\alpha}}\mathrm dx$

$$\ln^2(1+x)\sim x^2-x^3,x\rightarrow \infty\Rightarrow \int_1^{+\infty} \frac{\ln^2(1+x)}{x^{2\alpha}}\mathrm dx=\int_1^{+\infty} \frac{x^2-x^3}{x^{2\alpha}}\mathrm dx=$$ $$\int_1^{+\infty} ...
2
votes
0answers
63 views

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll ...
3
votes
1answer
72 views

How to calculate Lebesgue integral in this type?

For Lebesgue integral in this type, like$$\int_\pi^\infty \left({1 \over {x \sin^{1/3}x} }\right)^2$$ can anyone give me some general idea? I don't know use which inequalities to start with the this ...
4
votes
1answer
88 views

Is this equality $\lim_{x \to \infty} \int_0^x \frac{t^2}{2(e^t-1)}\mathrm{d}t=\lim_{n \to \infty}\sum_{k=1}^n \frac{1}{k^3}$ true?

Using a little program in Python, it looks true for at least two hundred digits after the comma, but I have absolutely no idea, how to begin. Any hint sould be appreciate. $$\lim_{x \to \infty} ...
3
votes
3answers
107 views

Question on reasoning for $\int_1^\infty\frac{\sin(x)}{x}dx$ to converge

I often saw a 'proof' that $\int_1^\infty\frac{\sin(x)}{x}dx$ converges: By integration by parts we get $$\int_1^\infty\frac{\sin(x)}{x}dx = \cos(1)-\int_1^\infty{\frac{\cos(x)}{x^2}}dx$$ and thus ...
0
votes
1answer
33 views

Solving $\iint\frac{x^2}{(8x^2+6y^2)^{\frac 3 2}}$ on the domain $8x^2+6y^2\leq 1$

I need to solve $\displaystyle\iint\frac{x^2}{(8x^2+6y^2)^{\frac 3 2}}$ on the domain $8x^2+6y^2\leq 1$. I recognise this is an improper integral, so we need a monotonic series of domains ...
1
vote
2answers
31 views

Confusion in the usage/property of Laplace Transform.

While proving that $$\int^{\infty}_0 \frac{\sin x}xdx$$ I saw the Laplace Transform proof. It used that $$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0 \cal L\left\{\sin(t)\right\}d\sigma$$ So ...
4
votes
1answer
52 views

Convergence of improper integral with $f(x)\to 1$ as $x\to +\infty$

Suppose $f\in \mathscr{R}$ on $[0,A]$ for all $A<\infty$, and $f(x)\to 1$ as $x\to +\infty$. Prove that $$\lim \limits_{t\to 0}t\int_{0}^{\infty}e^{-tx}f(x)dx=1 \quad (t>0).$$ Proof: Let's ...
3
votes
3answers
223 views

Solving $\int_{0}^{\infty} \frac{x^2}{x^8+5}dx$? [closed]

$$\int_{0}^{\infty} \frac{x^2}{x^8+5}\ \mathrm{d}x?$$ It's very hard for me to solve above integral. Please help me.