Integrals Definition and 1000 Threads

Hello. I'm studying improper integrals in real analysis. However, two problems are very difficult to me. If you are OK, please help me.(heart) 1.2. I have solutions about above problems. However, I don't know how I approach and find the way for solving them.

Homework Statement I'm trying to prove these two theorems a) if ## 0 \leq f(x) \leq g(x) ## for all x ## \geq 0 ## and ## \int_0^\infty g ## converges, then ## \int_0^\infty f ## converges b) if ## \int_0^\infty |f| ## converges then ## \int_0^\infty f ## converges. Obviously assuming...

$(a)\;\;:: \displaystyle \int\frac{1}{\left(x+\sqrt{x\cdot (x+1)}\right)^2}dx$ $(b)\;\;::\displaystyle \int\frac{1}{(x^4-1)^2}dx$ My Trial :: (a) $\displaystyle \int\frac{1}{(x+\sqrt{x\cdot (x+1)})^2}dx$ $\displaystyle \int\frac{1}{x\left(\sqrt{x}+\sqrt{x+1}\right)^2}dx =...

Given, \sigma_{b} = \vec{P}\bullet\hat{n} Now, integrate both sides over a closed surface, \oint \sigma_{b} da = \oint (\vec{P}\bullet\hat{n}) da My math is fuzzy, and I don't really understand this next step. \oint \sigma_{b} da = \oint \vec{P} \bullet d\vec{a}...

Consider the Gaussian Integral (eqn 2.64).. is anyone able to explain how the constant of normalization is rationalised?

Homework Statement Essential Mathematical Methods for the Physics Sciences Problem 15.7 Show that if f(z) has a simple zero at z0 then 1/ f(z) has a residue of 1/f'(z0). Then use this information to evaluate: ∫ sinθ/(a- sinθ) dθ, where the integral goes from -∏ to ∏. Homework...

Homework Statement If \displaystyle \int_0^1 \dfrac{e^t}{t+1} dt = a then \displaystyle \int_{b-1}^b \dfrac{e^{-t}}{t-b-1} dt is equal to Homework Equations The Attempt at a Solution I used the definite integral property in the second integral \displaystyle \int_{b-1}^b...

Ok, so I'd like some advice on doing integrals that involve a laplacian and a tensor for example =\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial x^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho}) where...

Homework Statement . Calculate by a line integral the following double integral: ##\iint\limits_D (y^{2}e^{xy}-x^{2}e^{xy})dxdy##, D being the unit disk. The attempt at a solution. Well, if we consider C to be the curve that encloses the region D (C is the unit circle), then C is a...

Sorry, the title doesn't match up 100% with the content of the topic, but that's because I've decided to be a little bit more explicit about my question. I am trying to walk through the proof of Euler's Equation from Calculus of Variations, and I'm a little bit confused by the final step...

again, i need some help here guys.$\displaystyle\int\frac{3x-1}{2x^2+2x+3}dx$ =$\displaystyle\int\frac{3x-1}{2\left[\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\right]}dx$ $\displaystyle a=\frac{\sqrt{5}}{2}$; $\displaystyle u=x+\frac{1}{2}$; $\displaystyle du=dx$; $\displaystyle x=u-\frac{1}{2}$...

i was thinking hard how to integrate this, but none of the techniques I know did work. please kindly help with this matter. thanks! $\displaystyle\int\sqrt{1+\cos\theta}d\theta$

Hi! I've got a problem with an integral. Let's assume we've got something like this: ∫R3d3x1∫R3d3x2∫R3d3x3∫R3d3x4P(|x1|)P(|x3|)δ(x1+x2)δ(x3+x4)W(|x1+x2|)W(|x3+x4|) xi is a vector The "δ" is the Dirac delta. P(|x|i) & W(|xi+xj|) are some functions I would like to make it looks a bit...

i'm kind of unsure of my solution here please check. $\displaystyle\int \sin^3x\cos^3x= \int(\sin x \cos x)^3 =\int(\frac{1}{2}\sin2x)^3=\frac{1}{8}\int \sin^3 2x dx$ $\displaystyle u=2x$; $\displaystyle du=2dx$; $\displaystyle dx=\frac{1}{2}du$ $\displaystyle \frac{1}{8}\int...

Homework Statement Using the integral ∫dx/1+x^2 = pi/2 from 0 to infinity as a guide, introduce a parameter and then differentiate with respect to this parameter to evaluate the integral ∫dx/(x^2+a^2)^3 from 0 to infinity Homework Equations The Attempt at a Solution ∫(1/1+x^2) =...

Quantum Phys Homework: I am given a function: $$f(x)=\frac{1}{10}(10-x)^2\,;\,0\leq{x}\leq{10}$$ and $$f(x)=0$$ for all other \(x\). I need to find the average value of \(x\) where $$\bar{x}=\frac{\int_{-\infty}^{\infty}x\,f(x)\,dx}{\int_{-\infty}^{\infty}f(x)\,dx}$$ I am not really even sure...

Please refer to attached material. For the first question, I have tried looking at examples and have noted that the bounds have been provided in a manner: like |z|=1 (as given in part ii) I am not sure how to get transform the given |z-pi|=pi in such a format, although i suspect it would be...

For a few of you, this probably isn't very challenging. But I'm going to post it anyways since I find it interesting. Show that for $0 \le \theta \le \pi$, $ \displaystyle \int_{0}^{\theta} \ln(\sin x) \ dx = - \theta \ln 2 - \frac{1}{2} \sum_{n=1}^{\infty} \frac{\sin (2n \theta)}{n^{2}}$.Also...

This is not so much a tutorial, but rather a collection of useful results and techniques. Some of the proofs will be quite long, since as much as possible, I'll aim to prove most results and functional relations used herein, rather than just present another's identity as fact. There will be a...

Homework Statement I have a question can the average value for an integral be negative. I don't see why not just checking. You know this evalutation f_ave = (1/b-a) ∫ f(x) dx Homework Equations thx The Attempt at a Solution

Within certain branches of analysis - both real and complex - the Inverse Tangent Integral (and its generalizations) can be quite useful. Similarly, it's much less well-known (= uglier? lol) cousin, the Inverse Sine Integral can be used to solve many problems. To that end, this is not really a...

Here are a few Vardi-type integrals I recently posted on another forum (some of you might have seen them)...Assuming the following classic result - due to Vardi - holds...\int_{\pi/4}^{\pi/2}\log\log(\tan x)\,dx=\frac{\pi}{2}\log\left[\sqrt{2\pi}\frac{\Gamma(3/4)}{\Gamma(1/4)}\right]Prove that...

gauss's theorem is also applicable to charge in motion.but how the surface integral has to be taken??

Problem: Let ##\vec{F}## be a vector function defined on a curve C. Let ##|\vec{F}|## be bounded, say, ##|\vec{F}| ≤ M## on C, where ##M## is some positive number. Show that ##|\int\limits_C\ \vec{F} \cdot d\vec{r}| ≤ ML ## (L=Length of C).Attempt at a Solution: I honestly have no idea where...

I hope this makes my question clear... suppose we have a triple integral of dzdydx for [0<x<1 , sqt(x)<y<1 , 0<z<1-y] and from the sketch we can see that 0<y<1 and 0<z<1... my question is this, if we change the integration to dzdxdy we get [0<x<y^2 , 0<y<1 , 0<z<1-y], is that the only way? or...

Hello all, I have a couple of questions. First, about the mean value theorem for integrals. I don't get it. The theorem say that if f(x) is continuous in [a,b] then there exist a point c in [a,b] such that \[\int_{a}^{b}f(x)dx=f(c)\cdot (b-a)\] Now, I understand what it means (I think), but...

Homework Statement ##\int_0^\infty \frac{a}{a^2+x^2} dx## Homework Equations All the basic integration techniques. The Attempt at a Solution So, I saw this problem and wanted to try it using a different method then substitution, which can obviously solve it pretty easy. Since it is a very...

Hi all, I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't. Is the following true? If F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n}...

1. Homework Statement [/b] Use the direct comparison test to show that the following are convergent: (a)\int_1^∞ \frac{cos x\,dx}{x^2} I don't know how to choose a smaller function that converges similar to the one above. The main problem is i don't know where to start. A simple...

Hello All. I'm currently in a crash course on X-ray Diffraction and Scattering Theory, and I've reached a point where I have to learn about Bessel Functions, and how they can be used as solutions to integrals of certain functions which have no solution. Or at least, that's as much as I...

I've been studying for a test and have been powering through the recommended problems and have stumbled upon a problem I just can't seem to figure out. $$\int_{0}^{\infty} \frac{logx}{1+x^{2}} dx$$ (Complex Variables, 2nd edition by Stephen D. Fisher; Exercise 17, Section 2.6; pg. 167)...

Hello I'd first like to state I know how to solve and I know the answer to this integral however when I first looked at the integral my initial thought was that it was equal to zero. I'd like to explain why I thought it was equal to 0 and hopefully someone can tell me where I went wrong. I...

I was looking at an example where it was evaluating a closed loop integral of a vector field around a triangle (0,0) (1,1) (2,0) by using greens theorem. This example was in the green's theorem section of the book so green's theorem must be used. Anyways the double integral was set up as follows...

I just want to verify is this the way to calculate the result of a definite integral with the given interval. Say the result of the integral over [0,##\frac{\pi}{2}##] is \sin(\theta)\cos(\theta)d\theta|_0^{\frac{\pi}{2}} [SIZE="5"]It should be...

And your boundaries are defined as: 0 < x < y < 1 How do you know the relationship between x and beyond this? That is, we know that y is between x and 1, but x is between 0 and y. We have a loop. In a specific example, I know the answer is, where f(x,y) = 8xy ∫∫8xy dx dy With bounds 0 to...

Good evening Im starting to learn quantum mechanics from Griffith's book however I am having problems when dealing with Gaussian integrals in the first chapter. What book should I read in order to understand this subject? are there resources about gaussian integrals out there? Thanks a lot.

Evaluate integrals: $$1) \int\frac{dx}{x^2\sqrt{x^2+4}}$$ $$2) \int\frac{dx}{2+2\text{sin}x+\text{cos}x}$$

Homework Statement Using double integrals, calculate the volume of the solid bound by the ellipsoid: x²/a² + y²/b² + z²/c² = 1 2. Relevant data must be done using double integrals The Attempt at a Solution i simply can't find a way to solve this by double integrals, i did with triple...

[SIZE="4"]lim_{x\rightarrow + ∞} \frac{\int^{x^3}_{0} e^{t^2}dt}{x \int^{x^2}_{0} e^{t^2}dt} Attempt at a solution: I don't really know where to start. Any hints?

Homework Statement Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 47 - 5x2 - 5y2 and the plane z = 2. Homework Equations x2 + y2 = r2 x = rcosθ y = rsinθ The Attempt at a Solution I substituted the z = 2 into the equation given, 2 = 47 -...

Hello users, I would like to know when do you use pattern recognition over integrals Someone told me it was that For example the integral below I would like to know the procedure to rewrite the numerators as (2x-2) + 3 Where does the 3 come from? I would really appreciate Thanks in...

Use the comparison test to find out whether or not the following improper integral exist(converge)? integral(upper bound:infinity lower bound:2) 1/(1-x^2) dx Here's my solution for 3),but I think something went wrong For all x>=2 0<=-(2-2x)<=-(1-x^2) that means: 0<=-1/(1-x^2)<=-1/(2-2x)...

Homework Statement ##\int_{2}^{\infty} ue^{-u} du## The Attempt at a Solution What I did was find the family of functions described by the indefinite integral ##\int ue^{-u} du## then found the limit as b increases without bound. $$=\lim_{b\rightarrow \infty}...

Recently in the 'Challenge Forum' the following integral has been proposed... $$\int_{0}^{\infty} \frac{\ln x}{x^{2}+ a^{2}}\ d x\ (1)$$ Scope of this note is to illustrate a general procedure to engage integrals like (1) in elementary way, i.e. without use comnplex analysis tecniques. The...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

so this year I've finshed limits , derivatives (that's it in cacylus)and i'd like to study integrals , ididn't study logarithms yet so idk if that's necessary, thanks

Do you need to know how to graph in order to establish which limit of an improper integrals is going to infinity?? for example: ∫tan(3x)dx from 0 to ∏/6 The integral diverges,, but how do you figure which constant you should use In this problem they put it as the limit a b aproaches...

Hi guys! Looking at the wiki page for abelian integrals I get no intuition on these scary monsters, & since I'm still not 100% ready as regards all the material in the chapters preceeding sections on abelian integrals in the reference books mentioned on that page I'd think I'd have problems...

Hi, I was wondering whether if ∫f×g dμ=∫h×g dμ for all integrable functions g implies that f = h? Thanks

Integrate (z^3-6z^2+4)dz where the function is any curve joining -1+i to 1. Z is complex number