Integration Definition and 1000 Threads

Homework Statement Integrate $$\int_0^1 dw \frac{w^{\epsilon+1} \ln((r+1-w)/r)}{1+r(1+w)}$$ for ##\epsilon## not necessarily an integer but positive and r is negative (<-1). The argument of the log is positive. Homework Equations Integration by parts The Attempt at a Solution [/B] I can...

Homework Statement integrate (4x+3)/(x^2+4x+5)^2 Homework EquationsThe Attempt at a Solution I know to solve this problem you have to work with partial fractions, in the solution we were given they solve as followed 4x+3=A(x^2+4x+5)'+B I don't know why they take the derivative of x^2+4x+5...

Homework Statement Suppose that the characteristic equation to a second order, linear, homogeneous differential equation with constant coefficients yielded two complex roots: \begin{array}{l} {\lambda _1} = a + bi\\ {\lambda _2} = a - bi \end{array} This would yield a general solution of: y =...

Homework Statement I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear, Homework Equations w refers to wannier function, which relates to the Bloch function ##\mathbf{R}## is this case...

Homework Statement Prove that if f'(x) = g'(x) for all x in an interval (a,b) then f-g is constant on (a,b) then f-g is constant on (a,b) that is f(x) = g(x) + C Homework Equations Let C be a constant Let D be a constant The Attempt at a Solution f(x) = antiderivative(f'(x)) = f(x) + C g(x)=...

im working on some homework that the instructor recently gave us the solutions to and I am confused on something that he's done. intailly i have ∫N2(x+a/2)2(x-a/2)2 the integral is from a/2 to -a/2 the next step is this ∫2N2(x2-a2/4)2 integrated from a/2 to 0 i don't understand why the...

Homework Statement I am working on a mass transfer problem and have this equation: d/dt(Vc1) = Ak[c1(sat) - c1]Homework Equations Initial conditions: x=0, c1=0 The Attempt at a Solution I know that the result from integration should be: c1/c1(sat) = 1 – e-(kA/V)t But I don't understand...

1. Given the function ##xy+cos y+6xy^2=0## , it follows that ## dy/dx=-y/x-siny+12xy##2. My problem is how do we integrate this derivative ## dy/dx=-y/x-siny+12xy## to get back the original function3.## ∫dy/dx dx=y ##

I need to get a few things straight about the integration operation (as an intro calc student). I understand that integration is a process that takes a function and returns its antiderivative. We can think of it as an operator, where ##\displaystyle \int...dx## is kind of like an opening and a...

\int_{-0.4088}^{-\infty}\,e^{-1/2.4^2}d struggling to solve this calculation. Not sure if i have written the formula in the right way. First post on this site. thanks

I'm not sure whether to put this here or in Linear Algebra, if any Mod feels it should go in Linear Algebra I won't mind. I've just been introduced to Fourier Series decompositions in my Linear Algebra text, and I understand all the core concepts so far from the Linear Algebra side of it (a...

Hi Pls can anyone explain how the attached picture was worked out? Thanks

In this thread, ramparts asked how integration by parts could be used in general relativity. suppose you have ##\int_M (\nabla^a \nabla_a f) g .Vol## Can it be written like ##\int_M (\nabla^a \nabla_a g) f .Vol## plus a boundary integration term (by integrating twice by parts)? I think thay it...

Homework Statement Given X,Y,Z are 3 N(1,1) random variables, (1) Find E[ XY | Y + Z = 1] Homework EquationsThe Attempt at a Solution I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side and...

Hey guys, need your help and hope someone takes the time: I need to solve the double integral by changing the order of Integration. I would really appreciate if you could illustrate the way of how to compute the solution. Best Estelle :)

. Homework Statement Q1. Let f(x) be any continuous function that satisfies: $$-2x≤xg(x)≤2x$$ for $$0≤x≤1$$ Find the upper and lower bounds for: $$\int_{0}^{1} \sqrt{1+g(x)+x^2}dx$$ Q2. Let h(x) be any continuous function that satisfies: $$-4≤h(x)≤x^2-4$$ for $$0≤x≤1$$ Find the upper and...

Homework Statement rewrite using the order dx dy dz \int_0^2 \int_{2x}^4\int_0^{sqrt(y^2-4x^2)}dz dy dx The Attempt at a Solution I am having trouble because i don't know what the full 3 dimensional region looks like but the part on the xy plane is a triangle bounded by x = 0 , y = 4 and y =...

In this paper we have p18 an integral on space time M. The author takes a 3 dimensional space like Cauchy surface ##\Sigma## which separates M in two regions, the future and the past of ##\Sigma##. He gets so the sum of two integrals on these regions. He writes then let us integrate each of them...

Homework Statement E = { (x,y) | |x| + |y| ≤ 1} fx,y (x,y) = { c (x,y) ∈ E 0 otherwise } Find C. Find the Marginal PDFs Find the conditional X given Y=y, where -1 ≤ y ≤ 1. Are X and Y independent. Homework Equations I'm taking a guess here in the solution... but F(x,y) = F(x)F(y) and...

I am having some problems to use the gsl library. I do the integration via monte carlo and I have obtained the right result, but a need to know how to use this result when I have a dependence on another variable. For example, f(x,y)=x*y, but the integration is only on x. Because I want to...

Mod note: Removed the color formatting, and surrounded the code with [ code ] tags. @whatisreality, please use a tag at the beginning of your code, and a tag at the end. Inserting colors by hand is distracting and makes your code difficult to read. I have to write a piece of code to calculate...

Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...

I only really need help evaluating the contour integral. I added more detail for completeness. 1. Homework Statement A particle of mass ##m## is constrained to move on the ##x## axis subject to the following potential. V(x)=a \sec^2\left( \frac{x}{l} \right) Find the action angle variables...

A force field is maintained around point O, a particle with mass m is experiencing a force F in the force field. F as a function of the particle's distance from O is: F = cos(d/5) How does one go about looking for the final velocity of the particle if it began at rest at a negligible distance...

Hello, This is not a homework problem, but a worked example I encountered from Hobson and Riley 3e pg 60, if anyone has the book. If not I took a screen shot. [I actually just decided to post the photos on a blog so no one has to download anything]...

I've been trying to teach myself the path integral formulation of quantum field theory and there's a point that's really bugging me: why is the integration measure ##\mathcal{D}\phi(x)## invariant under shifts in the field of the form $$\phi(x)\rightarrow\tilde{\phi}(x)=\phi(x)+\int...

Homework Statement (This is a part of the entire problem. I'm just struggling with going to the next step since it involves solving this integral.) Integrate: $$ \int \frac {1}{\sin \theta \sqrt {R^2\sin ^2 \theta - a^2} } d\theta $$ Homework Equations R and a are simply constants. Only $$...

This is not only a question strictly about mathematics, but in science or any other quantitative field in which there is an integration - or a summation that is like a discrete integration. [ A ] the parameter that is considered the input variable for the integration/summati - i.e., the x of dx...

In the example in the picture, we can see that they chose the limits of integration to be from 0 to R_0. Why didn't they choose x (that is, from 0 to x)? Isn't that what we normally integrate over when we find potential energy and electric fields? Thank you

Hallo Everybody, I am searching for a book (or lecture notes) that details the calculations that lead me from a given Lagrangian to the Feynman rules of the theory. It should not be rigouros, just the main steps to get the Feynman rules. Thanks for your help!

I am trying to understand how to go from the first line to the next: ##\frac{1}{(2\pi)^{3}}\int d^{3}p\ e^{-it\sqrt{{\bf{p}}^{2}+m^{2}}}\ e^{i {\bf{p}}({\bf{x}}-{\bf{x}}_{0})} = \frac{1}{2\pi^{2}|{\bf{x}}-{\bf{x}}_{0}|} \int_{0}^{\infty} dp\ p\ sin(p|{\bf{x}}-{\bf{x}}_{0}|)\...

So I was reading a solution of a problem that has to do with integration in polar coordinates, and in one of the steps it did this How did these terms (circled in blue) change like that? I've never seen a step like that before..

The Integral: \int{\sin{(\theta)}*\cos{(\theta)}*d\theta} Attempt to solve by Integration by Parts: \int{u*dv} = u*v - \int{v*du} u = \sin{(\theta)} du = \cos{(\theta)}*d\theta v = \sin{(\theta)} dv = \cos{(\theta)}*d\theta Bringing back to the beginning.

I'm struggling to solve the following integral ∫ x/(√27-6x-x2) my attempt is as follows: ∫x/(√36 - (x+3)2) = ∫1/ √(36 - (x+3)2) + ∫x+1/ √(36 - (x+3)2) = arcsin (x + 3)/6 + this is where I got stuck.

I want to know how this integral will equal zero? I know that Ψ will fall to zero as x goes to infinity and i know that Ψ must fall to zero very quickly , Ψ must fall to zero faster than 1/√|x| all of this will help evaluating this integral i tried to solve it as follows The first term...

How to find value of integration constant?I know with the help of boundary conditions,but How boundary conditions help in finding integration constant?

Hello, im sorry the picture is upside down. But my problem is with this formula: The book i am reading from says the net work done in moving a piston which must push gases, varying the pressure and volume is given by the formula shown in the picture. He then reduces it to what he calls a...

Homework Statement Evaluate \int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx via trigonometric substitution. You can do this via normal u-substitution but I'm unsure of how to evaluate via trigonometric substitution. Homework EquationsThe Attempt at a Solution Letting x=sinθ...

I have this question: Is it possible to define an orientation for a null submanifold with boundary? In that case, is possible to use Stokes' theorem? In particular, there is a way to define a volume form on that submanifold?

Homework Statement Homework Equations Doesn't the integral of sec x tan x equal sec x? The Attempt at a Solution

How to integrate $$\int_{-0.5}^{0.5} {\bigg({\frac{1}{4-2e^{-j2{\pi}f}}}\bigg)}^2 df$$

Homework Statement integrate tan^3(9x) Homework Equations tan^2(x)= sec^2(x)-1 integral of tanxdx= ln|secx| The Attempt at a Solution Integrate tan3(9x)dx Integral (sec2(x)-1)tan(9x)dx so then I distribute the tan(9x) giving me: integral tan(9x)sec2(x)dx - integral tan(9x) dx so then I solve...

My first homework for electrostatics course I'm taking is to find the vector of electric field of a completely hollow sphere (radius r, surface charge density σ in a point A, by integrating the electric field through the whole sphere. I already figured out the electric field of a ring in a point...

Homework Statement I am having trouble understanding the formula for Moments Of Inertia by direct integration. Homework Equations I understand the following (which is the definition) : $$ I_x = \int y^2 dA $$ $$I_y = \int x^2 dA $$ However come to application on a problem, my book doesn't...

Hi, I feel sometimes when I'm doing calculus I lose the logic and intuition behind what I'm doing, especially when integrating. I have yet to find a way to think about it in a way it makes sense to me why the definite integral would tell us the area under a curve. Same with why the second...

Homework Statement Prove that $$\int_{-\pi/2}^{\pi/2}\frac{log(1+b\sin x)}{\sin x}dx=\pi \arcsin(b)$$ Where ##|b|\le1## Homework Equations $$\frac{d}{dx}\big(\int_a^bf(t)dt\big)=\int_a^b\frac{\partial f(t)}{\partial x}dx$$ The Attempt at a Solution Let...

I've just found the following example on Piskunov: \int \frac{t^2 dt}{(t^2 + 2)^2}\\= \frac{1}{2}\int \frac{t d(t^2 +2)}{(t^2 + 2)^2}\\=-\frac{1}{2}\int t d(\frac{1}{t^2 + 2})\\=-\frac{1}{2}\frac{t}{t^2 + 2}+\int \frac{dt}{t^2 + 2}\\=-\frac{t}{2(t^2 + 2)} + \frac{1}{2...

So here is the problem: Find the anti-derivative of sec 3x(sec(3x) + tan(3x)) Now I have tried foiling it out, and I am stuck at the part where I need to anti-derive Sec(3x)Tan(3x). Any help/tips would be greatly appreciated.

Hello there, I'm reading a section of my textbook that is deriving the virial theorm from the hydrostatic equilibrium equation. In part of the derivation it states that $$-\int_0^M\frac{Gm(r)}{r}dm(r)=E_{GR}=-\frac{GmM}{r}$$ When I perform this integral I get the wrong answer. Here's my...

Hello there, I was wondering if someone might be able to help me out with some intermediate steps please. I can't see how $$f_{tt} =-\int_{r_0}^0 \left(\frac{2Gm_0}{r} - \frac{2Gm_0}{r_0}\right)^{-\frac{1}{2}} dr$$ becomes $$f_{tt} =-...