Integral Definition and 1000 Threads

Where , rho 1 and rho 2 are two dimensional position vectors and K is a two dimensional vector in the Fourier domain. I encountered the above Eq. (27) in an article and the author claimed that after integration the right hand side gives the following result: I tried to solve this integral but...

Homework Statement Let n be the unit outward normal of a spherical surface of Radius R, let the surface of the sphere be denoted by S. Evalute Surface integral of nndS Homework EquationsThe Attempt at a Solution I have evaluated the surface integral of ndS and found it to be 0. but am not...

After series of algebraic simplifications, I ended up with the following integral: ##\int_0^\infty \exp(-Kx) \arctan(x) dx ## As far as I searched, there is no closed form solution for the integral. But, K is my design variable that I need to optimize later. To do this, I need to take K out of...

Hi All, $$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$ I can perform the integration of the integral above easily by changing the variable $$u=x_2+x_1\\ v=x_2-x_1$$ Of course first computing the Jacobian, and integrating over ##u## and ##v## I am wondering how you perform the change of...

I am trying to evaluate the integral ##\displaystyle \int \frac{x}{1+\cos^2x}dx##. I have started by multiplying both the numerator and the denominator by ##\frac{\sin^2x}{\cos^4x}##, to get ##\displaystyle \int \frac{x\frac{\sin^2x}{\cos^4x}}{1+\tan^2x}dx##, and the denominator simplifies to...

Homework Statement [/B] Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is: Pn = Pn-1-√Pn-1 Homework Equations [/B] Considering that the...

When taking the superposition of wavefunctions with definite values of any observable (I'll just use momentum, but I am assuming it would work for any variable), I have seen the integral be used: ##\psi = \int_{-\infty}^{\infty}\phi(k)e^{ikx}dk## and the sum be used: ##\psi =...

Hello. I am having a lot of trouble trying to solve/analyse this integral: $$\displaystyle \int_2^\infty \frac{x+y}{(y)(y^2-1)(\ln(x+y))} dy$$ I have tried everything with no result; it seems impossible for me to work with that natural logarithn. I have also tried to compute it, as it...

Homework Statement A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by $\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda...

I have to find functions that maximise certain criterea. The problem can however not be put "under a single integral", for example I've to find ##f(t)##, ##g(t)## that maximise: ## \int_0^{t_e}f(t)^2dt\int_0^{t_e}g(t)^2dt - (\int_0^{t_e}f(t)g(t)dt)^2 ## With ## -1 \leq f(t)\leq1## and ## -1...

Homework Statement This is a combination of two questions, one being the continuation of the other 3) Calculate the DFT of the sequence of measurements \begin{equation*} \{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \} \end{equation*} 4a) Draw the DFT calculated in question 3 on the complex plane. 4b)...

Hi all, I was trying to find an answer, but couldn't, what is the integral of the squared probability density function? It doesn't seem to be equal to the square of cumulative distribution function, but how to tackle it? ∫(f(x))2dx = ? Can we transform it into, say, ∫f(x)dF(x)? and then...

Consider a system with a time-dependent Hamiltonian. We know that the evolution of the state of this system, is given by ## \displaystyle |\Psi(t_1)\rangle=T \exp\left( -i \int_{t_0}^{t_1} dt H(t) \right) |\Psi(t_0)\rangle ##. Do you think you can prove that the path integral formula for the...

Consider the following: $$\int \left(\frac{x-1}{x+1}\right)^4\,dx$$ I am unable to solve this.

Homework Statement How do I go about integrating ##(\frac{1-x}{1+x})^{\frac{1}{2}} ##? Homework Equations above The Attempt at a Solution im not really sure. could integrate by partial fractions if it was to the power of ##1##, only thing i can think of thanks in advance

Homework Statement Please don't make me post the entire question. If so, can I take a picture of the example in my textbook? I am looking at an example in my textbook where we are to check Stoke's theorem After doing the cross-product of del cross v I get (4z^2-2x)[x hat] + (2z^2)[z hat]...

So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...

This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me). The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...

Let V be the region bounded by the hemisphere z=1-sqrt(1-x^2-y^2) and the plane z=1, and let S be the surface enclosing V. consider the vector field $F= x(z-1)\hat{\imath}+y(z-1)\hat{\jmath}-xy\hat{k}$.

calculate $\displaystyle \int sze^z dS$ where S is the protion of the unit sphere centered at the origin such that x,y <0, z>0.

Online it says that the integral is the opposite of the derivative. So x^2 is the integral of 2x. So if f(x) = x^2 , does that mean that the integral is just the function itself? Basically whatever f(x) equals? Thanks in advance

Homework Statement please see attached, I am stuck on the second inequality. Homework Equations attached The Attempt at a Solution I have no idea where the ##2/\pi## has come from, I'm guessing it is a bound on ##sin \theta ## for ##\theta## between ##\pi/4## and ##0## ? I know ##sin...

In the path integral formalism, where we treat a photon as if it takes every possible path, aren't the possible paths limited by the speed of light? If we were to perform the double slit experiment, and shield the detector after a specified time frame to limit the time for a photon to make the...

Hello! Could you tell me about how to take the next numerical calculation in mathematica? (perhaps there are special packages). I have an expression (in reality slightly more complex): ## V=x^2 + \int_a^b x \sqrt{x^2-m^2} \left(\text Log \left(e^{-\left(\beta...

Homework Statement I'd like to know how to convert Maxwell's Equations from Differencial form to Integral form. Homework Equations Gauss' Law Gauss' Law for Magnetism Faraday's Law The Ampere-Maxwell Law The Attempt at a Solution Convert using properties of vector analysis (as Divergence and...

Could somebody write me the intuition behind the concept of "Integral Over"? Please do not write me its formal definition, I can easily get it from textbook. What I am also looking for is its motivation behind it. Please give me also examples. For your convenience, the formal definition...

Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...

The problem I am trying to show that the following integral is convergent $$ \int^{\infty}_{2} \frac{1}{\sqrt{x^3-1}} \ dx $$The attempt ## x^3 - 1 \approx x^3 ## for ##x \rightarrow \infty##. Since ## x^3 -1 < x^3 ## there is this relation: ##\frac{1}{\sqrt{x^3-1}} > \frac{1}{\sqrt{x^3}}##...

The problem $$ \int \frac{x}{\sqrt{x^2+2x+10}} \ dx $$ The attempt ## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx## Is there any smart substitution I can make here to make this a bit easier to solve?

The problem I want to find ##x## which solves ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt = 0 ## The attempt ##\int^x_1 \frac{\sin t}{t} \ dt = x -1 ## I see that the answer is ##x=1## but I want to be able to calculate it mechanically in case if I get similar problem with other elements. Any...

The problem Show that ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt < 0## for ## x > 1 ## The attempt I rewrite the integral as ##\int^x_1 \frac{\sin t}{t} \ dt < x-1 ## This is about where I get. Can someone give any suggestions on how to continue from here?

The problem I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty## The attempt ## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow...

The problem I want to calculate ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx ## for the step function below.The attempt I started with rewriting the function as with the help of long-division ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx = \int^6_{-6} 1 \ dx - 2\int^6_{-6} \frac{1}{g(x)+2} \ dx## I know...

Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.

I was trying to solve a problem involving work , as we know : w = \int_{a}^{b} \vec{f}.d\vec{s} but in my problem the path was cyrcular , so how to evaluate this kind of integral ?

Homework Statement Is the solution provided by the author wrong ? Stokes theorem is used to calculate the line integral of vector filed , am i right ? Homework EquationsThe Attempt at a Solution To find the surface integral of many different planes in a solid , we need to use Gauss theorem ...

$\textsf{evaluate}$ \begin{align} \displaystyle {I}&={\int{\frac{x+2}{x^2+1}dx}}\\ &=\int{\frac{x}{x^{2}{+1}}dx{\ +\ 2}\int{\frac{1}{x^{2}{+1}}}}{\ }dx\\ u&=x^{2}+1 \therefore \frac{1}{2x}du=dx\\ x&=\sqrt{u-1}\\ \end{align} ...? $\textit{calculator answer.?}$ $\dfrac{\ln\left(x^2+1\right)}{2}...

I am trying to find primitives to the rational function below but my answer differs from the answer in the book only slightly and now, I am asking for your help to find the error in my solution. This solution is long since I try to include all the steps in the process. The problem $$ \int...

Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...

I'm reading the path integral chapter of Schwartz's "Quantum Field theory and the Standard model". Something seems wrong! He starts by putting complete sets of states(field eigenstates) in between the vacuum to vacuum amplitude: ## \displaystyle \langle 0;t_f|0;t_i \rangle=\int D\Phi_1(x)\dots...

Homework Statement Find the green's function for y'' +4y' +3y = 0 with y(0)=y'(0)=0 and use it to solve y'' +4y' +3' =e^-2x Homework Equations ##y = \int_a^b G*f(z)dz## The Attempt at a Solution ##\lambda^2 + 4\lambda + 3 = 0 \to \lambda = -1,-3## ##G(x,z) = \left\{ \begin{array}{ll} Ae^{-x}...

Homework Statement Find the green's function for y'' +2y' +2y = 0 with boundary conditions y(0)=y'(0)=0 and use it to solve y'' + 2y' +2y = e^(-2x) Homework Equations ##y = \int_a^b G(x,z)f(z)dz## The Attempt at a Solution I'm going to rush through the first bit. If you need a specific step...

Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...

Consider the integral ##\displaystyle \int_{-\infty}^\infty \frac{e^{-|x|}}{1+x^2}dx ##. I should be able to use contour integration to solve it because it vanishes faster than ## \frac 1 x ## in the limit ## x \to \infty ## in the upper half plane. It has two poles at i and -i. If I use a...

Dear "General Math" Community, my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right...

Homework Statement Calculate ##\iint { y+{ z }^{ 2 }ds } ## where the surface is the upper part of a hemisphere with radius a centered at the origin with ##x\ge 0## Homework Equations Parameterizations: ##\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le...

Homework Statement I have calculate my double integral using wolfram alpha , but i get the ans = 312.5 , but according to the book , the ans is = 0 , which part of my working is wrong Homework EquationsThe Attempt at a Solution Or is it z =0 , ? i have tried z = 0 , but still didnt get the...

Homework Statement Homework Equations k∫[ƒ(x)]n ƒ'(x) dx The Attempt at a Solution i tried to using algebraic substitution to determine that i had let u = 1-x or X2-2x+1 or x or root(x) but it still cannot solve it. Please give me hint how to solve it. Thank you [/B]

I have: $$\int_{1}^{3} \frac{1}{\sqrt{3 - x}} \,dx$$ I can do $u = \sqrt{3 -x}$, and $du = \frac{1}{2 \sqrt{3 - x}} dx $ and $dx = 2 \sqrt{3 - x} du $. Plug into original equation: $$\int_{1}^{3} \frac{2 u }{u} \,du$$ and $2 \int_{1}^{3} \,du = 2u = 2 \sqrt{3 - x} + C$ So $(2\sqrt{3 - 3})...

I have: $$\int_{1}^{2} \frac{1}{x lnx} \,dx$$ I can set $u = lnx$, therefore $du = \frac{1}{x} dx$ and $xdu = dx$. Plug that into the original equation: $$\int_{1}^{2} \frac{x}{x u} \,du$$ Or $$\int_{1}^{2} \frac{1}{ u} \,du$$ Therefore: $ln |u | + C$ and $ln |lnx | + C$ So I need to...