Integral Definition and 1000 Threads

Homework Statement Show that ##\mathbb{Z} [\sqrt{d}] = \{a+b \sqrt{d} \ | \ a,b,d \in \mathbb{Z} \}## is an integral domain Homework EquationsThe Attempt at a Solution Do I have to go through all of the axioms to do this? For example, do I have to show that it is an abelian group under...

$\tiny{242.tr.05}$ Use the integral test to determine if a series converges. $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sqrt{e^{2n}-1}}$ so... $\displaystyle \int_{1}^{\infty} \frac{1}{\sqrt{e^{2n}-1}}\, dn =\int_{1}^{\infty} (e^{2n}-1)^{1/2} \, dn $ so $u=e^{2n}-1\therefore du=2e^{2n}$

Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...

I have this integral that when solved, involves squares and natural logs, where ##A\,##,##\,b\,##, and ##\,x_e\,## are constants while ##x## is a variable. ##A = \int_{x_e}^{x} \frac{x^2 - b^2}{x} dx = \int_{x_e}^{x} x \, dx -b^2 \int_{x_e}^{x} \frac{dx}{x} = \frac{x^2}{2} - \frac{x_e^2}{2} -...

Homework Statement Evaluate the indefinite integral as a power series. What is the radius of convergence (R)? ##\int x^2ln(1+x) \, dx## Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1## Homework Equations Geometric series ##\frac {1} {1-x} =...

Homework Statement Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of the square is a and one...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

Homework Statement Let ##R## be an integral domain and ##p_1(x),p_2(x) \in R[x]## with neither equal to ##0##. Show that the degree of ##p_1(x)p_2(x)## is the sum of the degrees of ##p_1(x)## and ##p_2(x)##. Homework EquationsThe Attempt at a Solution Here is my attempt. Let ##p_1(x) = a_n...

Homework Statement Calculate the integrals of the following functions on the given paths. Why does the choice of path change/not change each of the results? (c) f(z) = exp(z) / z(z − 3) https://www.physicsforums.com/file:///page1image10808 i. a circle of radius 4 centred at 0. ii. a circle...

Homework Statement Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving:: Homework Equations 3. The Attempt at a Solution [/B] As no...

I used wolframalpha to calculate an integral and here's what I got: ## \displaystyle \int \frac {dy}{y^d} \left( \frac 1 {\sqrt{1-y^{2d}}}-1\right)=\frac{y^{1-d}\left[ y^{2d} \ _2F_1\left( \frac 1 2,\frac{d+1}{2d};\frac {3d+1}{2d};y^{2d} \right)+(d+1)\left( \sqrt{1-y^{2d}}-1 \right)...

This is a chemically inspired problem, but the path is fully quantum mechanics and a bunch of integrals. How does one calculate fully quantum mechanical rate ($\kappa$) in the golden-rule approximation for two linear potential energy surfaces? Attempt: Miller (83) proposes...

Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...

Homework Statement ##\displaystyle \int_{- \infty}^{\infty} \frac{1}{\sqrt{2 \pi}} x e^{- \frac{x^2}{2}} dx## Homework EquationsThe Attempt at a Solution So first off, obviously the answer is 0, because the integrand is odd and we have symmetrical limits of integration. However, when I make...

Does anyone know how I can prove the following equation? ##\displaystyle \frac 1 {d-1}-\int_0^1 \frac{dy}{y^d} \left( \frac 1 {\sqrt{1-y^{2d}} }-1\right)=-\frac{\sqrt \pi \ \Gamma(\frac{1-d}{2d})}{2d \ \Gamma(\frac 1 {2d})} ## Thanks

Homework Statement Calculate the indefinite integral of the function ## \int\frac{3x^3}{\sqrt{1-x^2}}## my book gives the answer ##-(2+x^2)\sqrt{1-x^2}+C## Homework EquationsThe Attempt at a Solution So I started trying to calculate this indefinite integral by using a substitution...

Say we have the following result: ##\displaystyle \int_0^{\infty} \frac{\log (x)}{1 - bx + x^2} = 0##. We see that the denominator is 0 for some positive real number when ##b \ge 2##. Thus, we obtain a two singularities under that condition. Here's my question. Can we go ahead and say that the...

What is the transformation used Is there any explanation for : $$ \frac{\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{t0}\limits^{t}{{\mathrm{(}}{t}\mathrm{{-}}{s}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}}{x}_{0}\mathrm{(}s\mathrm{)}ds $$ How did become like this...

Say we have the following integral: ##\displaystyle \int_0^1 \frac{\log (x+1)}{x^2+1}##. I know how to do this integral with a tangent substitution. However, I saw another method, which was by differentiating ##f## under the integral with respect to the parameter ##t##, where we let...

Just a couple questions. Problem 2: Just would like to know if this is the correct approach for this problem. Problem 3: I am just wondering if I can use Problem 2 to prove the first part of Problem 3? Because to me, they seem very similar. Problem 4: Would I use the MVT for integrals...

I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?

Homework Statement This has been driving me crazy I can't for the life of me figure out how to convert the limits of this integral into spherical coordinates because there is an absolute value in the limits and I'm absolutely clueless as to what to do with with.Homework Equations $$\int_{\frac...

This is a heat equation related math problem. 1. Homework Statement The complete question is: Verify the orthogonality integral by direct integration. It will be necessary to use the equation that defines the λ_n: κ*λ_n*cos(λ_n*a) + h*sin(λ_n*a)=0. Homework Equations κ*λ_n*cos(λ_n*a) +...

Hi the parameter $\lambda$ in linear integral equations say Fredholm that appears in front of the integral containing the kernel i.e. $y(x)=f(x)+ \lambda \int_{a}^{b} \,k(x,t) y(t) dt $ can $\lambda$ be adjusted for the convergence of the method like in using Picard's successive...

∫x2√(3+2x-x2) dx Here's what I've already done: completed the square ∫x2√(4-(x-1)2) dx (x-1) = 2sinθ sinθ = (x-1)/2 x = 2sinθ+1 dx = 2cosθ dθ trig sub + pulled out constants 4∫(2sinθ+1)2√(1-sin2θ)cosθ dθ trig identity 4∫(2sinθ+1)2√(cos2θ)cosθ dθ 4∫(2sinθ+1)2(cos2θ)dθ expanded + trig...

Homework Statement Evaluate ##\displaystyle \int_{1}^{\infty} \frac{dx}{(x+a)\sqrt{x-1}}## Homework EquationsThe Attempt at a Solution First I make the substitution ##u = \sqrt{x-1}##, which ends up giving me ##\displaystyle \int_{0}^{\infty} \frac{2u}{u(u^2 + 1 + a)}du##. Here is where I am...

Homework Statement Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ The Attempt at a Solution If the interpretation of an integral is the derivative at that point, x1 for example, then these can't all be solutions since the tangent, the derivatives...

Homework Statement Hi. Can anyone here solve this integral for me: ##\int\frac{d^3y}{\operatorname dx^3}\;y\;dx## Homework Equations I haven't seen such integral before, it is included in an old math book and it should be solved using "Integration by Parts" technique because it is an exercise...

Homework Statement Let ##R## be a principal ideal domain and suppose ##I_1,I_2,...## are ideals of ##R## with ## I_1 \subseteq I_2 \subseteq I_3 \subseteq ...## The Question has two parts: 1. to show that ##\cup _{i=0}^{\infty}I_i## is an ideal. 2. to show that any ascending as above must...

Homework Statement f(xy)=49/8*e^(−3.5*y) 0 < y < inf and −y < x < y 0 otherwise a. Find the marginal probability density function of X, fX(x). Enter a formula in the first box, and a number for the second and the third box corresponding to the range of x. Use * for multiplication, / for...

Homework Statement Suggest an integral that is reduced to a rational function integral when this substitution is used: ##a)## ##t=\sin x## ##b)## ##t=\sqrt[6] {x+5}## ##c## ##\sqrt{1-9x^2}=-1+xt## Homework Equations 3. The Attempt at a Solution [/B] I found this to be a very interesting...

Homework Statement Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ $$F=ma,~a=\frac{dv}{dt}$$ The Attempt at a Solution $$-\frac{mgR}{s^2}=mv\left( \frac{dv}{ds} \right)\rightarrow~v^2=\frac{2gR}{s}+C$$ $$v_0(s=R)=\sqrt{2gR}\rightarrow~C=2g(R-1)$$...

Homework Statement I'm trying to find the value of the integration: Homework Equations Mod note: Edited the TeX to render properly at this site. Here is the integration written using MAketex: ##\frac{.5}{\sqrt{\pi}}\int_0^{\infty}\exp(-z(1+1.5/v))z^{L-.5} \frac{1}{(\frac{z}{v}+1)^n...

stevendaryl submitted a new PF Insights post Solve Integrals Involving Tangent and Secant with This One Weird Trick Continue reading the Original PF Insights Post.

Homework Statement Hello, I've recently encountered this double integral $$\int_0^1 dv \int_0^1 dw \frac{(vw)^n(1-v)^m}{(1-vw)^\alpha} $$ with ## \Re(n),\Re(m) \geq 0## and ##\alpha = 1,2,3##. Homework Equations I use Table of Integrals, Series and Products by Gradshteyn & Ryzhik as a...

Homework Statement Find f(x) if:(a∈R) $$f(x)=\int_0^{\frac{\pi}{a}}f(t)cos(at-2ax)dt+1$$ Homework Equations $$f(x)=\int_{a}^{b}f(t)dt=F(a)-F(b)$$ $$\int{udv}=uv-\int{vdu}$$ The Attempt at a Solution I tried to use the fundamental of calculus and integration by parts but they don't have answer...

1. Homework Statement $$\frac{dy}{dx}=\sqrt[3]{\frac{y}{x}},~x>0$$ Why do i need the x>0, indeed my result is good for all x since it contains x2 2. Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ 3. The Attempt at a Solution $$\int...

$$\left \int \frac{dy}{y^{1/3}} \right.$$ Doesn't work. There also was an examples page in your site, i don't find it

Homework Statement Hi, I am not asking for solution for any problem as i already have the given solution for the problem. Instead, what i want clarify is what do they mean by the odd and even function and how do they get 0? Also, is there a need to change the order from dxdy to dydx?

Hello! Can someone suggest me a good reading about path integral formulation of quantum mechanics? I took 2 undergrad courses on QM, so I would like something focusing on path integral (maybe some problems too). I don't necessary want a book, even some online pdf that contains some good...

Homework Statement The problem is in the attached file. The part I need a little help with is part b. Homework Equations and attempt at a solution[/B] For part a, I got h(8) = 2, h'(6) = -2, and h''(4) = -2. For part c, I found that the integral from 0 to 5 is 7, so I multiplied 7 by 7 to...

Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...

Hello, I have tried the integral below with Mathematica and it gives me the following solution: ##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}## I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...

Homework Statement I can find the e-field at point P. Homework Equations I get, easily enough, the correct integral solution (for the y-component, Ey - which is all I need to do): which I can see, informally, evaluates to: which is the correct answer. The Attempt at a Solution My...

I would like to prove that the following integral is logarithmically divergent. $$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$ This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##...

Hello! First time poster, please treat me well! :wink: I've already solved the problem below on my second attempt with the help of kinetic energy but I want to know why my first attempt gives a wrong answer. 1. Homework Statement A force in the +x-direction with magnitude F(x) = 18.0 N -...

Joseph A. Gallian, in his book, "Contemporary Abstract Algebra" (Fifth Edition) defines an irreducible element in a domain as follows ... (he also defines associates and primes but I'm focused on irreducibles) ... I am trying to get a good sense of this definition ... My questions are as...

Joseph A. Gallian, in his book, "Contemporary Abstract Algebra" (Fifth Edition) defines an irreducible element in a domain as follows ... (he also defines associates and primes but I'm focused on irreducibles) ... I am trying to get a good sense of this definition ... My questions are as...

I have read many textbooks and googled google times for a clear explanation, but I could not find one. How does raising and lowering -annihilation/ creation-(is that energy or particle number?) translate to transition probabilities of path integral.