Integral Definition and 1000 Threads

Hey! :o A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$. I want to calculate $\int_0^{T_0}x^2(t)\, dt$. I have done the following: It holds that...

We would need to recognise that the integral in the equation is a convolution integral, which has Laplace Transform: $\displaystyle \mathcal{L}\,\left\{ \int_0^t{ f\left( u \right) \,g\left( t - u \right) \,\mathrm{d}u } \right\} = F\left( s \right) \,G\left( s \right) $. In this case...

Hi, I have pasted two improper integrals. The text has evaluated these integrals and come up with answers. I wanted to know how these integrals have been evaluated and what is the process to do so. Integral 1 Now the 1st integral is again integrated Now the text accompanying the integration...

Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} Consider the expression: \begin{align} \int_{q'}^q \int_{b'}^b g(s)\ \int_{s-t}^{s+t} f(r)\ dr\ ds\ dt\ \end{align} We...

Hi, the above image is from the Line Integral Convolution paper by Cabral and Leedom. However, I am having a hard time implementing it, and I am quite certain I am misreading it. It is supposed to give me the distances of the lines like in the example below, but I am not sure how it can. First...

I solved the integral by two different methods and I get different answers. Method 1: ∫dx/(1-x) = -∫-dx/(1-x), u=1-x, du=-dx ∫dx/(1-x) = -∫du/u = -ln|u| = -ln|1-x| Method 2: ∫-dx/(x-1) = -∫dx/(x-1), u=x-1, du=dx ∫-dx/(x-1) = -∫du/u = -ln|u| = -ln|x-1| What am I doing wrong?

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Let's say you have a tensor u with the following components: $$u_{ij}=\nabla_i\nabla_j\int_{r'}G(r,r')g(r')dr'$$ Where G is a Green function, and g is just a normal well behaved function. My question is what is the square of this component? is it...

Thanks

I'm trying to solve the inequality: $$ \int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx $$I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there. Any ideas?

So the problem I’m attempting to solve is ##\lim_{x\to a} I_{\alpha}f(x)=\zeta (\alpha )## for f, and a, where ##\zeta (\cdot )## is the Riemann zeta function and ##I_{\alpha}## is the Riemann-Liouville left fractional integral operator, namely the integral equation $$\lim_{x\to...

This question hopefully isn't going to go too deep into the concept, just a couple of questions to get me going. I am working on using dimensinal regularization of a loop integral in QED. I don't think the specific application to QED is important, but I will say that the original integral is...

Homework Statement:: I need to develop my instincts on when to use u-sub, integration-by-parts, trig substitution, etc. But, I need to read/see tons of problems actually being solved with these techniques to know which technique to apply quickly. Relevant Equations:: Sorry for the vague...

Part of me thinks this is could be a u-sub b/c x^3's derivative is 3x^2, a factor of 3 off from what e is raised to...but it is not a traditional u-sub...any thoughts if this is a u-sub or by parts, and what u should be?I know that there is more to solving the equation after this ( z =...

Let ## E=\left\{ (x,y,z) \in R^3 : 1 \leq x^2+y^2+z^2 \leq 4, 3x^2+3y^2-z^2\leq 0, z\geq0 \right\} ## - Represent the region E in 3-dimensions -represent the section of e in (x,z) plane -compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz## the domain is a sphere of radius 2 with an inner...

I have tried integration by parts where, ##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*} = \frac{1}{\sqrt{(r*)^3}} \frac{r^{3/2} dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##u = r^{3/2} \quad \quad dv = \frac{dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##du = \frac{3}{2} r^{1/2} dr...

I have a few questions and a request for an explanation. I worked this problem for a quite a while last night. I posted it here. https://math.stackexchange.com/questions/3547225/help-with-trig-sub-integral/3547229#3547229 The original problem is in the top left. Sorry that the negative...

Hello, I found an integral to calculate the torque from the applied torsional shear stress, and I didn't find an explanation of how this integral is deviated. Where does it come from? Could someone explain? T = ∫τ⋅r⋅dA = ∫τ⋅2πr⋅dr, where T is the torque and τ the shear stress. Thanks a lot!

This could be solved by the substitution ##u=\sqrt x##, but I wanted to do it using a trigonometric one. The answer is false, but I don't see the wrong step. Thank you for your time! [Poster has been reminded to learn to post their work using LaTeX]

are the boundaries of integration correct? i split the domain in two as follows -2<=x<=0 , -(4-x^2)^(1/2)<y<=x+2 and 0<=x<=2 -(4-x^2)^(1/2)<=y<=(4-x^2)^(1/2)

I want to understand where the minus 1 in the first line in the RHS term comes from. I assume the little apostrophe means taking a derivative. But the antiderivative of x^(n-1) is (1/n)x^n. Why the -1? thank you

I was just playing with the integral ##\int e^{ixa}dx## when I found something interesting. If you integrate from ##x = m2\pi/a## to ##x = n2\pi/a## where ##m## and ##n## are any two integers, the integral equals zero. On one hand, as we can in principle choose whatever values we like for ##m##...

find out for what values of p > 0 this integral is convergent ##\displaystyle{\int_0^\infty x^{p-1}e^{-x}\,dx}\;## so i broke them up to 2 integrals one from 0 to 1 and the other from 1 to ∞ and use the limit convergence test. but i found out that there are no vaules of p that makes both of...

The Lerch Transcendent identity from my paper which may or may not be true, for ##N\in\mathbb{Z}^+##, and I forget the domain of z and y, here it goes $$\Phi (z,N,y) :=\sum_{q=0}^{\infty}\frac{z^q}{(q+y)^N}$$ $$=\int_{0}^{1}\int_{0}^{1}\cdots \int_{0}^{1}\prod_{k=1}^{N}\left(...

The equations here come from calculating the amplitude of a Feynman diagram. I can set up the problem if you really want me to but here I am just interested in why and how the regularization process is supposed to work Mathematically. The generalized meaning of this is if we are given a...

D={(x,y)∈ℝ2: 2|y|-2≤|x|≤½|y|+1} I am struggling on finding the domain of such function my attempt : first system \begin{cases} x≥2y-2\\ -x≥2y-2\\ x≥-2y-2\\ -x≥-2y-2 \end{cases} second system \begin{cases} x≤y/2+1\\ x≤-y/2+1\\ -x≤y/2+1\\ -x≤-y/2+1\\ \end{cases} i draw the graph and get the...

If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...

Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...

could please some one explain the inequality on the right? in particular how should i see and thanks

I think its going to be intg(dr2)intg(exp(r^2) dr) or something like that.

e.g Can we write it as $$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+...f(b)=\int^b_a f(x)dx$$...(?) Although $$\int f(x)dx$$ given the area tracked by thr function with the x-axis between a and b Thanks.

Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region

Hi everyone, I am confused in this question. First I solved it by noticing that the gradient of the function will be zero (without substitution the hit) I got that it's a conservative field so the integral should be zero since it's closed path. Then I solved it by the hit and convert it as any...

Hi everyone, I tried to solve the last part of the question, I substituted back the expression of x and y into the equation of the ellipse, I got that r=1 or r=-1. But got no idea how to find the boundary for theta, I got a guess that, It should be from zero to pi. But got no reason why to...

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 3: Integration ... and in particular I am focused on Section 3.1: The Riemann Integral ... I need help with an aspect of the proof of Proposition 3.1.4 ...Proposition 3.1.4 and its proof read as follows...

I tried to do this just by observation, but kinda hard with a piece wise function so would presume

Hello there, I'm struggling in this problem because i think i can't find the right ##\theta## or ##r## Here's my work: ##\pi/4\leq\theta\leq\pi/2## and ##0\leq r\leq 2\sin\theta## So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta## Which is equal to...

I can only find a solution to \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel...

Hello, How to find formulas for these$\displaystyle\int x^n\sin(x)\, dx, \displaystyle\int x^n\cos(x)\, dx,$ indefinite integrals when $n=1,2,3,4$ using differentiation under the integral sign starting with the formulas $$\displaystyle\int \cos(tx)\,dx = \frac{\sin(tx)}{t}...

So the task is to solve the following integral with laplace transform. Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)). What I am unsure about is what happens with the integral part and how do we inpret the resulting expression? What will it result...

I am now reading this paperhttps://arxiv.org/pdf/gr-qc/0405103.pdf, which is related to the energy condition in wormhole. Nevertheless, I got a problem in Eq.(6), which derives from so-called ANEC in Eq.(2): $$\int^{\lambda2}_{\lambda1}T_{ij}k^{i}k^{j}d\lambda$$ And I apply the worm hole space...

image due to macros in Overleaf ok I think (a) could just be done by observation by just adding up obvious areas but (b) and (c) are a litte ? sorry had to post this before the lab closes

15.1.34 Evaluate $\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}} \sin{y} \, dz \, dx \, dy$ integrat dz $\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy $ integrat dx $\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$...

As a simple example, the probability of measuring the position between x and x + dx is |\psi(x)|^{2} dx since |\psi(x)|^{2} is the probability density. So summing |\psi(x)|^{2} dx between any two points within the boundaries yields the required probability. The integral I'm confused about is...

The answer gives $$ \int x /(2x-1)\ dx = x/2 +(1/4)ln|2x-1| + C $$ whicjh I can obtain. But when I try a different way I get a different answer. I must be making a stupid mistake but I can't see it. Here is my method $$ \int x/(2x-1) dx = \int x/[2(x-(1/2)] dx = (1/2) \int x/(x-(1/2)) dx $$...

Using spherical coordinates I can write ##d^3 k = 2\pi k^2 \sin \phi dk d \phi## (where I've already preformed the integration along the azimuthal angle, yielding the factor ##2 \pi##). Btw I'm sorry for my unfortunate notation: usually ##\phi## is the azimuthal angle, but here it is the polar...

Some functions have straight foward integrals, but they get complicated if you take the inverse of it. 1/f(x) for instance. The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges. ##\int_1^\infty (\ln(x))^n dx## If n = 0, I...

In physics we often change a sum to an integral.But I am not clear when can we change a sum to an integral?When a term of sum is comparable to the sum,can we change the sum to integral?