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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11
votes
3answers
288 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First I try the substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
3
votes
0answers
43 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. When $|z|$ is large in magnitude and $-\pi < \arg(z) ...
1
vote
0answers
12 views

Prove that the condition $x(\tau)>\xi$ of a divergent integral implies that $x(t)>\xi$

Let $ E, J \subset \mathbb R$ be open intervals and let functions $h:J \to \mathbb R$ and $g: E \to \mathbb R$ be continuous. let $\xi \in E$ and assume that $g(\xi)=0$. Define $f:J \times E \to ...
7
votes
1answer
804 views

contour integration of a function with two branch points .

Many of us have seen the evaluation of the integral $$\int^{\infty}_0 \frac{dx}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$$ It can be solved using contour integration or beta function . I thought of ...
1
vote
1answer
21 views

absolutely integrability implies function approaches zero at positive infinity

Is the following statement true? $$\text{If function $f$ is absolutely integrable on $[0, \infty)$, this implies } \lim_{x \rightarrow \infty} f (x) = 0.$$ If yes then how would I prove it? Note: I ...
0
votes
3answers
36 views

Integral from $-\infty$ to $\infty$ of a function?

Integral from $-\infty$ to $\infty$ of $xe^{-x^2}$. Now I know if the integral from $a$ to $\infty$ of $f(x) dx$ and $-\infty$ to $a$ are convergent, then I could find the integral by summing the ...
25
votes
1answer
300 views
+500

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Let $\operatorname{li} x$ denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ...
1
vote
0answers
31 views

Is there a close form expression for the integral $ \int_a^b |x-c|^n e^{-x^2/2} $

Is there a close form expression for the integral \begin{align} \int_a^b |x-c|^n e^{-x^2/2} dx \end{align} by close form I mean it can be in terms of well know functions such as $Q$-function, ...
16
votes
8answers
702 views

A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$

How to prove the following $$\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx=\frac{\pi^2}{2}$$ I thought of separating the two integrals and use the beta or hypergeometric ...
-1
votes
1answer
17 views

how to prove if $f$ is integrable and $f'$ too then the limit of $f$ is zero when $x$ go to infinity? [on hold]

If $f$ is a real function on $\mathbb R$ and we have $\int_1^\infty |f(x)|dx < \infty$ and $\int_1^\infty |f'(x)|dx < \infty$ then $\lim_{x\to\infty}f(x)=0$ ?
-2
votes
2answers
95 views

Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$ [on hold]

As a consequence of this Q, I need some help evaluating the following integral: $$\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$$ Integration by parts wouldn't simplify things and I guess that ...
6
votes
1answer
105 views

Improper integral: $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $.

mathematica is reporting that the improper integral $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $ coverges to $2\cos(1)$. However, when I try to confirm this by actually integrating it using ...
1
vote
6answers
114 views

Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$ [duplicate]

I am familiar with the gauusian integral $$\int_{-\infty}^\infty e^{-\alpha x^2+\beta x}dx=\sqrt{\frac{\pi}{\alpha}}e^{\beta^2/(4\alpha)}$$ Could anyone help me to find out the value of the following? ...
1
vote
2answers
51 views

To test convergence of improper integral $\int_{0}^{1} \left(\log\left(\frac{1}{x}\right)\right)^m\,\mathrm dx$

To test convergence of improper integral $$\int_{0}^{1} \left(\log\left(\frac{1}{x}\right)\right)^m\,\mathrm dx$$ I made cases and I am stuck on case in which I have to check convergence for ...
1
vote
0answers
35 views

Convergence of improper integral $\int_{2}^{\infty} \frac{1}{log(t)}dt$ [duplicate]

Convergence of improper integral $\int_{2}^{\infty} \frac{1}{log(t)}dt$ How do i start?
0
votes
0answers
24 views

Testing convergence of a sum with two improper integral in it

For wich $\alpha$ does the series converge?$$\sum_{n=3}^{\infty}\sin{\{2\pi n^2 + [\int_{(\ln n)^\alpha}^\infty arctan(t)*(\sin {1/t})^3 dt]*[\int_{\ln n}^\infty \frac{arctan(t^2)}{e^t +2}dt]\}}$$ i ...
0
votes
2answers
58 views

To test convergence of improper integral $ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\, \mathrm dx$

I have to test convergence of improper integral $$ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\,\mathrm dx$$ I write as $\log(x) \leq x$ . So $x\log(x) \leq x^2$. So $ \frac{x\log(x)}{(1+x^2)^2} ...
2
votes
3answers
93 views

Evaluating $\int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$

The problem is to evaluate the improper integral: $I = \int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$. This can be written as follows: $$I = \int_0^{\infty} dt \frac{\sin(xt)}t \int_0^a ...
1
vote
2answers
80 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
2
votes
2answers
69 views

Compute integral: $\int_0^{+\infty}\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-\frac{1}{2} (x^2+y^2)}dx dy $

I would like to resolve this exercise: Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$. I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ ...
5
votes
2answers
153 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
2
votes
1answer
407 views

Calculate integral using beta and gamma functions

I have to calculate the following integral using beta and gamma functions: $$ \int\limits_0^1 \frac{x\,dx}{(2-x)\cdot \sqrt[3]{x^2(1-x)}} $$ I came up with this terrible solution. Firstly, let's ...
2
votes
1answer
74 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
1
vote
1answer
64 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
2
votes
0answers
48 views

Sine improper integral

Suppose the following integral $$ \int\limits_{-\infty}^{\infty}\sin{x}dx $$ In mathematical rigor, the following is the definition $$ \int\limits_{-\infty}^{\infty}\sin{x}dx = ...
5
votes
2answers
435 views

$\int_{0}^{\infty} f(x) \,dx$ exists. Then $\lim_{x\rightarrow \infty} f(x) $ must exist and is $0$. A rigorous proof?

Let $f: \mathbb R \rightarrow \mathbb R $ be a continuous function such that $\int_{0}^{\infty} \,f(x) dx$ exists. Then Prove that incase (i) $f$ is a non negative function, then ...
2
votes
2answers
50 views

How to show the convergence of the integral $\int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t}t^x\mathrm{d}t$?

The integral is defined, for all $x\in\mathbb{R}$ as follows: $$I= \int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t} t^x\mathrm{d}t.$$ When $I$ converges? Let $t-1=u$, we have: $u\to 0$ when $t\to 1$. ...
0
votes
1answer
53 views

How do I evaluate $\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\lambda}} dt$ ??

How do you evaluate: $$\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\alpha}} dt = ??$$ Many thanks.
0
votes
0answers
27 views

Convergence of improper integral with parameter

In my assignment I have to study the convergence of this integral: $$\int_{0}^{1} \frac{ln(1 + \sqrt{x})}{x (x^{\alpha}-1)} dx$$ with the parameter $\alpha >0$. In a neighbourhood of $x=0$ I ...
3
votes
1answer
52 views

Evaluating $\int_0^{\infty} \frac{\sin xt \sin yt \cos zt}{t^2} \, dt$

The problem is to evaluate the improper integral $I = \int_0^{\infty} \frac{\sin xt \sin yt \cos zt}{t^2} dt$. This can be written as $\int_0^{\infty} dt \int_0^y \frac{\sin xt \cos st \cos zt}{t} ...
2
votes
1answer
32 views

Solve the improper integral: $\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$

I'm completely stuck on this one. I only know that it converges thanks to Wolfram, but I don't know how to evaluate it. $$\int_1^{\infty}\frac{33e^{-\sqrt{x}}}{\sqrt{x}}$$ Thank you for the help.
1
vote
1answer
21 views

When to stop simplifying an improper integral

When evaluating whether an improper integral is convergent or divergent, I'm sometimes unsure whether I simplified enough to be sure. For example, given $$ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sec ...
3
votes
0answers
50 views

How can revolving an infinite area have a finite volume [duplicate]

The area of the region bounded by $f(x) = \frac{1}{x}$, $y = 0$, and $x = 1$ is $$ A = \int_1^{+\infty} f(x) \, \textrm{d}x = \lim_{b \to +\infty} \int_1^b \frac{\textrm{d}x}{x} = \lim_{b \to ...
0
votes
2answers
68 views

Showing convergence of the integral $\int_{1}^{\infty}\frac{\ln(x)\cos(x)}{x^2+1}\,{\rm d}x$

I need to show that the following integral either converges or diverges. $$\int_{1}^{\infty}\frac{\ln(x)\cos(x)}{x^2+1}\,{\rm d}x$$ I am fairly certain it converges, but am stuck on showing how. ...
0
votes
2answers
55 views

Find $\int_0^\infty \frac{\sin(4x)}{x}$

How would one go about computing $$\int_0^\infty \frac{\sin(4x)}{x}$$ without any background in complex analysis (e.g. using strictly calculus)? I know that $$\int_0^\infty \frac{\sin(x)}{x} = ...
4
votes
3answers
176 views

Does the integral $\int_{a}^{b}\frac{dx}{\sqrt{(x-a)(x-b)}}$ exist?

What is the result of this integral $\displaystyle\int_{a}^{b}\dfrac{dx}{\sqrt{(x-a)(x-b)}}$ ? I have tried many possibilities like letting $\sqrt{(x-a)(x-b)}$=u or trying to make the denominator ...
2
votes
2answers
121 views

Find the exact value of $\int^\infty_0\frac{5x^{2/3}}{-2-3x^{12/5}} dx$

Find the exact value of $$\int^\infty_0\frac{5x^\frac{2}{3}}{-2-3x^\frac{12}{5}} dx$$ I've tried using integration by parts. If you make $U=x^{10/15}$ then you end up with a more complicated ...
0
votes
2answers
40 views
2
votes
3answers
88 views

How to see this improper integral diverges?

$$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$ I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?
0
votes
0answers
35 views

Finding an integral for an expression

I have the equation below. If I work backwards and integrate the second line w.r.t. t and then evaluate at t = x, I can get the first line. However, how do I go from the first expression to the ...
1
vote
2answers
105 views

Evaluate $\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$

This is a integration question from a previous calculus exam: Evaluate $$\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$$ I rewrote it as $$\lim \limits_{b \to \infty} \int_0^b ...
-1
votes
0answers
29 views

double integral with Gamma function

How can I solve the integral $$\int_{0}^{\infty}\int_{\delta t}^{\infty} e^{-x}x^{\beta-1}dxd\delta$$ or how can I check weather the integral is proper or not. Can anyone help me to reduce it into ...
0
votes
1answer
35 views

Generalisation of an already generalised integral

Inspired by these two questions: Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$ Interesting integral formula I ask whether the following integral has a closed form: ...
1
vote
2answers
41 views

For which real $a$ is the integral $\int_1^\infty x^ae^{-x^3\sin^2x}dx$ finite?

For which $a \in \mathbb{R}$ is the integral $\int_1^\infty x^ae^{-x^3\sin^2x}dx$ finite? I've been struggling with this question. Obviously when $a<-1$ the integral converges, but I have no idea ...
13
votes
1answer
559 views

Two integrals involving logarithm and polylogarithm function

Evaluate the following integrals $$\int_0^1\frac{\ln(1-x)}{x}\text{Li}_3\left(\frac{1 + x}{2} \right)\,dx\\ .\\ \int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1 + x}{2} \right)\,dx$$
0
votes
0answers
18 views

Solutions to Heat equation $ \int_{\mathbb{-\infty}}^{\infty} \frac{\partial^2 T(x, t)}{dt^2} dx = 0 $

I was wondering about the motion of heat and came across this differential equation. $$ \int_{\mathbb{-\infty}}^{\infty} \frac{\partial^2 T(x, t)}{dt^2} dx = 0 $$ $T(x,t)$ represents temperature ...
3
votes
3answers
61 views

Show that $\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}}$ converges.

Show that $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} $$ converges. I recognized that that since the integrand is even then $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} = ...
2
votes
1answer
49 views

What is the nature of this improper integral?

Consider this improper integral of first kind: $$\int_0^{+\infty}{\frac{t\ln t}{{(t^2+1)}^{\alpha}}}\,{dt}, \quad \alpha\in\mathbb R$$ Its required to find the nature of this improper integral. We ...
1
vote
4answers
87 views

How to solve this integral by parts?

I was solving a problem of mean values, and I would like to solve and evaluate this integral: $$ \langle x^2\rangle=\int_{-\infty}^{\infty}\left(\frac{2\alpha}{\pi}\right)^{1/2}x^2 ...
0
votes
3answers
53 views

Evaluate $\int_{2}^{+\infty} \frac{1}{x \log^2{x}} dx$

I'm having some difficulties solving this improper integral: $$\int_{2}^{+\infty} \frac{1}{x \log^2{x}} dx.$$ Taking the limit as $b$ approaches infinity we have $$\lim_{b\to\infty}\int_{2}^{b} ...