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
3answers
122 views

A closed form of $\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
0
votes
2answers
32 views

Convergence of improper integral of $\ln f(x)$

Is there something know about the convergence of $\int_0^1 \ln f(x)dx $ for $f(x)$ continous on $\left(0,1\right)$ and both limits exists, i.e. $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$ ? I ...
1
vote
1answer
41 views

Improper double integral evaluation by changing the order of integration

I was watching an MIT OCW recitation video about exchanging the order of integration on double integrals. The example was: ...
2
votes
1answer
86 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
2
votes
2answers
37 views

Fresnel Integral multiplied with cosine term.

$$I=\int_a^b \sin(\alpha-\beta x^2)\cos(x)\, dx.$$ Can anybody tell me, how to solve this integral ? I know that this is related to Fresnel Integral if the $\cos(x)$ term is absent.
5
votes
3answers
104 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
-1
votes
1answer
42 views

Value of line integral [on hold]

Let $C=\{(x,y)\in \Bbb R^2:~\max\{|x|,|y|\}=1\}.$ The value of the line integral $$\oint_C(xy^2+2y+sin(e^x))dx+(x^2y+cos(e^y))dy$$ is?
5
votes
5answers
177 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
1
vote
1answer
95 views

Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
2
votes
1answer
38 views

Improper parametric arc length

The first thought I had to solve this problem was using the integral, $$ \int_1^\infty \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\: \:dt $$ Once you solve for the derivatives ...
2
votes
2answers
37 views

Prove $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$

I try to evaluate following integral $\int_{-\infty}^{\infty} \frac{du}{(1+u^{2})^{\frac{7-p}{2}}} =\frac{\sqrt{\pi}\Gamma[\frac{1}{2}(6-p)]}{\Gamma[\frac{1}{2}(7-p)]}$ It seems okay to extend ...
1
vote
0answers
16 views

Approximate improper Bessel integral

In the classic book "Conduction of Heat in Solids" by Carslaw & Jaeger (1959) the heat diffusion equation $$ \frac{\partial T}{\partial t}=\alpha\left(\frac{\partial^2T}{\partial ...
9
votes
3answers
147 views

Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$

I have in trouble for evaluating following integral $$\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$ It seems really easy, but I ...
3
votes
0answers
29 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
3
votes
2answers
116 views

Find $\lambda$ if $\int^{\infty}_0 \frac{\log(1+x^2)}{(1+x^2)}dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}dx$

Problem : If $\displaystyle\int^\infty_0 \frac{\log(1+x^2)}{(1+x^2)}\,dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}\,dx$ then find the value of $\lambda$. I am not getting any clue how to proceed ...
1
vote
3answers
29 views

Growth restriction for nonnegative, continuous functions whose integrals on $\mathbb{R}$ are bounded

When we have a nonnegative, continuous function $f(x)$ whose integral over all real numbers $\mathbb{R}$ is bounded, like: $$\int_{-\infty}^{\infty}f(x)dx = A< \infty $$ with $A \in \mathbb{R}$ ...
1
vote
0answers
52 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
0
votes
1answer
27 views

How to identify continuity or discontinuity of an [Definite] integral?

How can I figure out whether an improper integral converges based on the discontinuities in the integrand? For instance, these two both have discontinuities within the intervals of integration, and ...
3
votes
1answer
36 views

Show $\iint xye^{-xy}\,dx\,dy$ is convergent or divergent

Determine convergence/divergence of $$\iint xye^{-xy}\,dx\,dy$$ for $x,y \geqslant 0$ i.e. in the first quadrant. I have managed to show that $xye^{-xy} \to 0$ in the first quadrant but other ...
4
votes
1answer
74 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
1
vote
1answer
57 views

Help finishing this exercise!

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $F$ is ...
3
votes
1answer
39 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
3
votes
1answer
33 views

Finding a dominating function for this sequence of functions

Problem: Find the limit $$\lim_{n\to\infty} \int_0^n \left( 1 + \frac{x}{n}\right )^{-n} \log(2 + \cos(x/n))dx$$ and justify your reasoning. My Solution: Let $f_n = \left( 1 + \frac{x}{n}\right ...
1
vote
0answers
152 views

$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$

The following integral bothers me since weeks: $$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$$ Has any body a suggestion for this integral. $Re\; s >0$ sufficiently large and $s'$ an ...
8
votes
1answer
99 views

Interesting sum-integral equality

Is there an elementary proof of $$\lim_{n \to \infty} \int_0^\infty e^{-\alpha x^2} \frac{\sin((2n + 1)x)}{\sin x} dx = \pi\left(\frac{1}{2} + \sum_{k = 1}^\infty e^{-\alpha k^2 \pi^2}\right),$$ where ...
0
votes
1answer
44 views

What is the limit of $nf(x+n)$ as $n\rightarrow \infty$? Here $f(x)$ is prob. density function.

I tried the cases when $f(x)$ are the densities of normal and student t distribution. In both cases, the limit is $0$. I guess this conclusion might hold in general. I tried the following. Let ...
1
vote
0answers
48 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
8
votes
1answer
255 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
1
vote
1answer
59 views

Proving $\int^\infty_0 x^n e^{-x} \, dx = n!$

I was motivated by this question on the various applications of integration by parts to prove the following integral: $$\int^\infty_0 x^n e^{-x} \, dx = n!$$ Here's what I have done, I feel I am ...
2
votes
1answer
26 views

fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
1
vote
1answer
49 views

Surface integrals with normal derivatives.

Define $G(x)= \frac{1}{4\pi ||x||}$, suppose that $f(x)$ is known, S is a surface in $\mathbb{R}^3$, and x is fixed, $x \in S$. I have formulas for computing the following numerically: $$ p(x) = ...
2
votes
0answers
53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
1
vote
2answers
74 views

Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
2
votes
4answers
136 views

Prove that $\lim_{n\rightarrow \infty} \int_{0 }^{\pi} \frac{\sin(nx)}{nx}dx=0$

Prove that $$\underset{n\rightarrow \infty }{\lim} \ \int_{\epsilon }^{\pi} \frac{\sin(nx)}{nx}dx=0\ ;\ \epsilon>0$$ then use the result to deduce: $$\underset{n\rightarrow \infty }{\lim} \ ...
-2
votes
0answers
56 views

Convolution Integral

Does someone know how to solve this convolution integral? $$V(x)=c_1\int\limits_{-\infty}^\infty \left(r(\tilde x)+\dfrac{c_2}{n(\tilde x)}-n(\tilde x)\right)\left(\sqrt{c_3+(x-\tilde x)^2}-|x-\tilde ...
0
votes
1answer
50 views

Evaluating integral involving Bessel function.

Evaluate $$\int_0^{\infty } \frac{2^{\frac{r}{\delta }} \left(2^{\frac{r}{\delta }}-1\right) r\ e^{-\frac{\alpha ^2+\left(2^{\frac{r}{\delta }}-1\right)^2}{2 \beta ^2}}\log 2 }{\beta ^2 \delta } ...
2
votes
1answer
74 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
1
vote
2answers
88 views

Limit of a function with a defined integral 3

Let $F$ be function defined as an integral $$F(x)=\int_{1}^{\infty}\dfrac{t^ke^{-xt}}{1+t^{5}}\textrm{d}t \quad \forall k\in \mathbb{N},\ x>0$$ Show that $\lim_{x\to \infty}F(x)=0$ ...
1
vote
0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
2
votes
2answers
110 views

Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
0
votes
0answers
29 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
2
votes
3answers
101 views

How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
3
votes
1answer
162 views

How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over ...
4
votes
3answers
262 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
5
votes
3answers
172 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
1
vote
1answer
41 views

Evaluating improper integral expression

Can anybody please guide me in evaluating this expression, for my research work. I have tried a lot but in vain. Both the integrals involve gamma functions and even wolfram said that the time limit ...
0
votes
2answers
36 views

Evaluating Improper integral

Using the equation $\frac{1}{\sqrt{x}}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\exp(-\alpha^{2}x) \, \mathrm{d}\alpha$, for $\alpha>0$, Compute the two integrals $$\int_{0}^{\infty} ...
1
vote
2answers
82 views

Prove that $\Gamma\left(t+1\right)=t\ \Gamma\left(t\right)\quad\forall{t>0}.$ [closed]

Consider the next function: $$\Gamma\left(t\right)=\int_{0}^{+\infty}x^{t-1}e^{-x}dx.$$ Prove that $\Gamma\left(t+1\right)=t\ \Gamma\left(t\right)\quad\forall{t>0}.$
0
votes
1answer
35 views

Convergence of improper integral depending on parameter [closed]

For what $a>0$ $(\frac{1}{\sin x})^a$ is integrable on $(0,\frac{\pi}{2})$?
2
votes
1answer
101 views

Proving $\displaystyle \int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$ Exists

As the title says, I need to show that $$\int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$$ exists. After performing the substitution $x = 1/u, dx = -1/u^2 du$, the integral becomes ...