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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1answer
42 views

Conditional convergence $\int_{-\infty}^{+\infty} \frac{(x-1)\sin 2x}{x^2-4x+5}dx$

Explore conditional convergence $$\int_{-\infty}^{+\infty} \frac{(x-1)\sin 2x}{x^2-4x+5}dx$$ I tried $$\int_{-\infty}^{+\infty} \frac{(x-1)\sin 2x}{x^2-4x+5}dx = \int_{-\infty}^{+\infty} \frac{\sin ...
3
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2answers
44 views

A sufficient condition for the existence of an improper integral (or a counterexample for it)

Let me try to explain the spirit of the question. The functions $f(x)=1/x^{p}$ for $0<p<1$ and $f(x)=\ln x$ have the following properties: they are in some respect 'nice' on $\mathbb{R}^{+}$, ...
2
votes
2answers
132 views

What is $\displaystyle\int_{2}^{2}\frac{dx}{x-2}$?

Evaluate the integral: $$\displaystyle\int_{2}^{2}\frac{dx}{x-2}.$$ 1)When does $\displaystyle\int_a^a f(x)dx=0$? Always? 2)Does $\displaystyle\int_a^a$ means area between $(a,a)=\emptyset$? 3) Do ...
1
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1answer
31 views

Surface Integral over a rhombus

Evaluate the integral $$\int\int_{R}(x-y)^2 cos^2(x+y)dxdy$$ where $R$ is the rhombus with successive vertices as $(\pi,0), (2\pi,\pi), (\pi,2\pi), (0,\pi).$ My attempt- I tried doing this surface ...
2
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3answers
42 views

Prove $\int\limits_1^\infty x^a\sin x \, dx$ diverges for $a>1$

Let $a>1$. I need to show that $$ \int_1^\infty x^a\sin x \, dx $$ diverges. I am not sure, but this is my progress We will look first at intervals $[2m\pi,(2m+2)\pi]$. Then $$ ...
1
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1answer
35 views

Integrating the bivariate normal distribution

Let $X$ and $Y$ have the bivariate normal density function, $$ f(x, y) = \frac{1}{2 \pi \sqrt{1 - p^2}} \exp \left\{ - \frac{1}{2(1 - p^2)} (x^2 - 2pxy + y^2) \right\} $$ for fixed $p \in (-1, 1)$. ...
0
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1answer
23 views

Method used for improper integrals can be applied to proper integrals also?

If $f$ is continuous on $[a,b]$, show that $$\lim_{c\to a^+}\int_{c}^{b}f(x)dx=\int_{a}^{b}f(x)dx$$ Hint: A continuous function on a closed finite interval is bounded and there exists a ...
1
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1answer
30 views

Improper Integral: Comparison Test

I have the following improper integral: $$\int ^\infty _{-\infty}\frac{2016}{e^x+e^{-x}} \, dx$$ My question is how to prove that it is convergent or divergent by using the Comparison Test.
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0answers
16 views

Prove improper integral converges

I'm studying the behaviour of the Bessel function as $x \rightarrow \infty$ part of the assignment requires me to prove the following: Prove the improper integrals $$\int_x^\infty ...
-1
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1answer
38 views

Find the value of p for which the integral converges and evaluate integral for $\int ^\infty_e \frac{1}{x(\ln x)^p} dx$

Find the value of p for which the integral converges and evaluate integral for $\int ^\infty_e \frac{1}{x(\ln x)^p} dx$ MY ATTEMPT: Given Integral: $I= \int ^\infty_e \frac{1}{x(\ln x)^p} dx$ put ...
0
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1answer
20 views

Integral of reciprocal of absolute value

I am having trouble with the following question. For which values of $n, p$ is the integral $\int_{\mathbb{R}^n}\frac{1}{|x|^p}$ convergent? I need to derive a relationship between $n$ and $p$, i.e. ...
0
votes
1answer
29 views

Would the sum after applying the integral test be equal to the sum of a series?

I know that in order to apply the integral test for convergence or divergence a function $f(x)$ must be positive, continuous, and decreasing. However, I was wondering if $$ \int_{1}^{\infty}f(x)\, ...
0
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0answers
41 views

Calculate non-elementary integrals

I'd like to calculate $\int_{-\infty}^{\infty}\sin(x^2)dx$ and $\int_{-\infty}^{\infty}\cos(x^2)dx$ I think it may be possible to do it by using the fact that: $$\int_{\gamma}^{} e^{-z^2}dz =0$$ For ...
1
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3answers
167 views

Integrate $\int_0^{2 \pi} \frac{2}{\cos^{6}(x) + \sin^{6}(x)} dx$

Integrate $$\int_0^{2 \pi} \frac{2}{\cos^{6}(x) + \sin^{6}(x)} dx$$ I have tried to do Tangent substitution, but there is huge power. What method is better to use here?
0
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1answer
46 views

Improper integral.

I have example which asks to determine whether the improper integral converges or diverges, and if it converges I have to solve it. I couldn't solve this, but I found a very descriptive solution, but ...
3
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2answers
30 views

difference between two improper integrals

I can't grasp the difference between $\int_{-\infty}^{\infty}\,f(t)\,dt$ and $\lim\limits_{x \to \infty} \int_{-x}^x\,f(t)\,dt$ for example if $ f(t)=t $ then the first one will give a divergent ...
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4answers
93 views

How to prove $\int_{0}^{\infty}xe^{-x}dx$ converges without calculating the limit

I am given $\int_{0}^{\infty}xe^{-x}dx$ and asked to prove that it converges and if it does, calculate the integral. I have calculated the integral and it gives 1. However, I cannot find a way to ...
5
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0answers
88 views

Complicated exponential integral

I encountered the following integral, which I am unable to solve: $$\int_0^\infty {x\over a} e^{{x \over a}} e^{{-b \over a}(e^{{x \over b}}-1)} e^{{-c \over d}(e^{{x \over c}}-1)} \mathrm{d}x$$ ...
2
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1answer
103 views

Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$

I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 < \beta$. I used substitution and integration to get $$ ...
0
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1answer
84 views

Can $\int_0^1\frac{1}{t}e^{-t} dt$ be analytically or numerically integrated?

The following integral has a singularity at $t = 0$ as in this situation the exponential term becomes $1$ and it no longer dominates the $\frac{1}{t}$ term. $$f(x) = \int_0^1\frac{1}{t}e^{-t}dt$$ So ...
2
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0answers
104 views

Calculation of a double integral [closed]

Can you help me to calculate this integral. $$\begin{align}I&=\int_{0}^{1}\frac{1}{x^2+1}\left(\int_{0}^{1}\frac{dy}{1+xy}\right)dx\\ &=\int_{0}^{1}\frac{ln(x+1)}{x(x^2+1)}dx\\ ...
2
votes
2answers
59 views

How to compute this integral without the use of the error function?

I was watching this: https://youtu.be/qQ-56b_LvOw?t=4484 And this integral came up. $$\int_{0}^{\infty}{(2x^2+1)e^{-x^2}}dx$$ To which the answer was $\sqrt{\pi}$. They made it clear that you ...
0
votes
1answer
43 views

How to show this sequence is a delta sequence?

Consider the sequence $(\phi_n)_{n\in \mathbb{N}}$ of test-functions $\phi_n\in \mathcal{D}(\mathbb{R})$ defined by $$\phi_n(x) = \dfrac{n}{\sqrt{\pi}}e^{-n^2x^2}.$$ I want to show that this is a ...
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3answers
126 views

Computing an improper integral involving polynomial multiplied by an exponential

in my calculus class we are currently dealing with improper integrals and I was tackled with the following: $$ \int_{0}^{\infty} x^4 e^{-x^2}dx = $$ and $$ \int_{0}^{\infty} x^5 e^{-x^2}dx = $$ I ...
1
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3answers
48 views

Difficult integral of exponential function

Is there a closed-form solution for integrals of the form $$ V(a,b,c) := \int_a^\infty \exp( - b \, x^c) \, \mathrm{d}x \quad a,b,c > 0 . $$ Only for the special case $c=1$ I can figure out ...
0
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1answer
34 views

How to compute this integral with contour integration?

Consider the function $$g(z)=\dfrac{e^{izt}\phi(z)}{z},$$ where $\phi$ is a $C^\infty$ function. I want to compute the integral $$I=\int_{-\infty}^{\infty}\dfrac{e^{ixt}\phi(x)}{x}dx,$$ where $t$ ...
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0answers
21 views

Conditional and absolute convergence of the integral depending on the parameter

How to determine values of $p$ making the integral $$\int_{x=2}^{\infty}\frac{(x+1)^{p}\sin(x)}{\log(x)}\mathrm{d}x$$ converges absolutely or conditionally? Comparison $\log(x)$ with $x$ where ...
0
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1answer
80 views

Evaluate a tricky trigonometric integral with unusual limits?

How do you calculate $$\int^{5\pi/2}_{0} \frac{dx}{2+\cos x}$$ I tried all available substitutions including tangent half angle, but all these substitutions do not distinguish between $\pi/2$ and ...
0
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1answer
13 views

On absolute and conditional convergence of trigonometric integral

I am trying to make out if $\int_1^{\infty}{\arctan(\frac{\cos(x)}{x^{2/3}})}dx$ converges absolutely or conditionally. My answer is that there is no convergence at all, because substitution ...
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2answers
26 views

Some uncommon improper integral. Convergence.

I bet the integral $\int_1^3\frac{dx}{\sqrt{\tan(x^3-7x^2+15x-9)}}$ converges, but have no idea about proof, except expansion of tan in a series near zeros of its argument (limits of the integration). ...
1
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1answer
24 views

On convergence/divergence of improper integral with hyperbolic function

I am trying to determine whether $\int_0^{\infty}{(\frac{1}{xsinh(x)}-\frac{1}{x})dx}$ converges or diverges. It seems like inevitably divergent in 0 point. But how to show it? Maybe it should be ...
2
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1answer
58 views

Simplifying Integral Expression w/ Antiderivative

I'm trying to simplify this formula here. For the most part, evaluating this expression is straightforward. However I am very confused about how to approach this last term in the expression. I want ...
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2answers
35 views

Improper Integrals Value of a Constant

I have the integral: $$\int ^\infty _0\left(\frac{1}{\sqrt[2]{x^2+4}}-\frac{\Phi}{x+2}\right)dx$$ Where $\Phi$ is a constant. The question is to find the value of this constant, so this integral ...
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0answers
20 views

Collocation method for integral equation with monotone increasing kernel

Is it possible to approximate the solution ($f(x)$) of this type of integral equation if the kernel ($k(x,t)$) is a strictly monotone increasing function? $f(x-T)= g(x)+\int_0^{\infty } k(x,t) f(t) ...
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0answers
37 views

Convergence of this integral related to Riemann-Zeta Function

Is it possible to show that for $Re(s)>0$ the following integral converges. $$\displaystyle\int_1^\infty \left[{x^{\frac{s}{2}-1} + x^{-\frac{s+1}{2}} }\right] \omega \left({x}\right)dx$$ Where ...
0
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1answer
27 views

Does this integral exist as improper Riemann integral?

$\int xdx$ as upper bound goes to infinity and lower bound goes to negative infinity. $\int xdx$ as the bounds go to finite b and -b exists and is always equal to 0. Does that mean that this integral ...
0
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1answer
34 views

Improper Integrals Comparison Method

I have the Integral: $$\int^\infty_{20}\frac{1}{x\cdot \ln^{15}(x)}\,dx$$ I know that $$\lim_{x\to \infty}(\ln(x)) = \infty$$ Subsequently, I could substitute with $$\ln(x)$$ in the denominator and ...
0
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1answer
33 views

Improper Integrals Convergent or Divergent

I have the integral: $$\int_1^\infty \frac{2}{x(1+\cos^2(x^2+x+1))}dx$$ I could not figure out how to represent the equation in the denominator, so I could apply the limit. I need only to find if it ...
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4answers
504 views

Fundamental Theorem of Calculus Confusion regarding atan

According to this site, $$ \int \frac{1}{a^2 \cos^2(x) + b^2 \sin^2(x)} \,dx =\frac{1}{ab} \arctan\left(\frac{b}{a} \tan(x)\right)$$ Thus, $$ \int_0^{\pi} \frac{1}{a^2 \cos^2(x) + b^2 \sin^2(x)} \, ...
1
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1answer
21 views

Divergence of an integral

I am sure this is a familiar example to many, as it comes up as an example of a function which belongs to $L^p$ for $p=1$ but not $L^p$ for $p < 1$, but I am having a hard time seeing why it is ...
1
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1answer
39 views

Name of a particular improper integral

I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
2
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3answers
212 views

Seemingly Obvious Improper Integral Property

The following seems extremely obvious, so much so that I cannot see how to prove it: If $f:[a,b]\to\mathbb{R}$ is Riemann integrable then $$\int_a^bf(x)dx=\lim_{c\to b^-}\int_a^cf(x)dx$$
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3answers
64 views

Evaluate $\displaystyle\int\limits_0^{\infty}\frac x{20}e^{-x/20} dx$

I tried doing $u$-substitution and got $-20e$ as my final answer, but I think the correct answer is just $20$. I'm not sure what I did wrong, but probably had to do with plugging in infinity... could ...
1
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1answer
39 views

Limit to find convergence of improper integral

Show the integral is convergent and find the value it converges to. $$\int_1^\infty \frac{\arctan x}{x^2}$$ I have found the indefinite integral to be $$-\frac{\arctan x}{x} + \ln|x| ...
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0answers
47 views

Why the integral converges?

Say we take one of the loveliest functions in mathematics the Gaussian which looks like this: Picture of Gaussian. By eye inspection we can say that this looks like something that could have finite ...
2
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0answers
33 views

Why do the integral and the partial sum agree for small $a$ and $m$?

Consider the following naive manipulations: \begin{align} \int_0^\infty \frac{e^{-x}}{1+ax}\:dx & = \int_0^\infty e^{-x}\frac{1}{1-(-ax)}\:dx\\ &= \int_0^\infty e^{-x} \left( ...
0
votes
2answers
31 views

Improper Integral of $(y-1)^{-3/2}$ from $0$ to $2$

improper integral $$\int_{0}^{2} \frac{1}{(y-1)^{3/2}}\, dy$$ I know it doesn't work when $y = 1$, so I split the integral, right. But then i realized, it doesn't work with $0$ either, as that would ...
1
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1answer
27 views

Laplace transform identity $F(s) = \mathcal{L}(t^{-3/2} \mathrm{e}^{-1/t})$

I'm asked to prove the following result: If $F(s)$ is the Laplace transform of $f(t) = t^{-3/2} \mathrm{e}^{-1/t}$, show that $F'(s)=-s^{-1/2}F(s)$. I'm having a lot of troubles to prove this ...
0
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0answers
24 views

Discussion on convergence of improper integrals

I have a general question about the convergence of improper integrals and it is this: If we have an improper integral that converges, is this similar or analogous at all to saying that a function is ...
0
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1answer
40 views

Improper convergence of this integral?

$$ \int_1^{\infty} \left\langle t\right\rangle\dfrac{\cos\left(t\right) - \sin\left(t\right)}{t^2}\,dt $$ where $\left\langle t\right\rangle$ is the rationale part of $t$. I would like to use the ...