Integral Definition and 1000 Threads

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?

Hi, I am trying to solve the problem in Griffith's book about variational principle. However, I am having trouble to solve the integral by myself that I have indicated in redbox in Griffith's book. You can see my effort in hand-written pages. I brought it to the final step I believe, but can't...

Hi, I find this... Please tell me your opinion on this. Thanks.

So I'm trying to figure out the integral of phi hat with respect to phi in cylindrical coordinates. My assumption was that the unit vector would just pass through my integral... is that correct? (I reached this point in life without ever thinking about how vectors go through integrals, and...

I was playing around with a graphing program and sketching polar graphs involving tall power towers, when I noticed that ##sin(\theta) \uparrow \uparrow a## has an alternating appearance depending on whether ##a## is odd or even. I also noticed that the area enclosed by these alternating graphs...

I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?

integral of x^12 sinx dx = x^12 -cosx - 12x^11 sinx - cosx (132x^11/11)

Starting from the general formula: $$I_{n,m}=\frac{1}{(4\pi)^2}\frac{\Gamma(m+2-\frac{\epsilon}{2})}{\Gamma(2-\frac{\epsilon}{2})\Gamma(n)}\frac{1}{\Delta^{n-m-2}}(\frac{4\pi M^2}{\Delta})^{\frac{\epsilon}{2}}\Gamma(n-m-2+\frac{\epsilon}{2})$$ I arrived to the following...

Before writing out each component I'm going to simplify ##\vec{I}## to the best of my abilities $$\vec{I} = \int \left(\hat{r}\cdot\vec{r'}\right) \vec{r'} \rho\left( \vec{r'} \right)\, d^3r'$$ $$\vec{I} = \hat{r} \cdot \int \vec{r'} \left( x' , y', z' \right) \rho\left( \vec{r'} \right)\...

Hi all, Having this equation derived: ##\int _0^{\frac{\pi }{2}}\:sin^{n}x\:dx\:=\:\frac{n-1}{n}\int _0^{\frac{\pi }{2}}\:sin^{n-2}x\:dx## What I will do is simply substitue n with n+2, and I will get the following: ##\frac{2n}{2n+1}\int_{0}^{\pi /2}(sinx)^{2n-1}dx## What should I do from here?

I have an integral: \int_{-1}^{0}\int_{-1}^{q}\delta(s+a)\sinh[k(q-s)]dsdq where 0<a<1 and \delta (s-a) is a dirac delta function. Anyone know what to do?

If I take the exponential function e^t and take the derivative, I think I get the same e^t. Even if I keep doing it over and over, second, third derivative, etc. My admittedly naive question, though, is this symmetric? Meaning...if I take the the integral of e^t, do I just get the reverse or...

Let's take an integral##−\int_1^e\frac{dx}{x}##. On one hand, this is equal to ##-\ln(x)|_1^e##. But on the other, ##−\int_1^e\frac{dx}{x}=\int_1^e\frac{dx}{-x}##. If I assume that the integral of this is ##\ln⁡(-x)|_1^e##, then I'd be really stupid since ##\ln## is not even defined over the...

Hello all, I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the...

7.4.32 Evaluate the integral $\displaystyle \int_0^1\dfrac{x}{x^2+4x+13}\, dx$ ok side work to complete the square $x^2+4x=-13$ add 4 to both sides $x^2+4x+4=-13+4$ simplify $(x+2)^2+9=0$ ok now whatW|A returned ≈0.03111

Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. ___________________________________________________________________________ Consider the following multiple integral: ##\displaystyle B= \iint_S \Biggl( \iiint_{V'}...

Summary: Ιntegral calculation : (sin(x))^4 * (cos(x))^6 Hi all, I tried to solve it, but I got stuck. An advice from my professor is to set: x=arctan(t) Τhanks.

Find integration of: \frac {1}{(x+1)(x^2 + x -1)^\frac{1}{2}} What I did: 1. Use completing square method for the term inside the square root 2. Use trigonometry substitution (I use secan) 3. After simplifying, use another trigonometry substitution (I use weierstrass substitution) 4. Use...

How to solve $$\int_{-\infty}^{\infty} \frac{e^{-iax}coth[sinh[bx]]}{sinh[bx]} dx$$ mathematica gives the result ::idiv: "Integral of E^(-Iax)\ Coth[Sinh[bx]]\ Csch[bx] does not converge on {-\[Infinity],\[Infinity]}." thanks!

I had been trying to aplly the sokhotski–plemelj theorem but with no success. Moreover i replaced exponential function with taylor expansion but i still can not solve the integral. thanks

I have to integrate this expression so I started to solve the delta part from the fact that when n=0 it results equals to 1. And the graph is continuous in segments I thought as the sumation of integers $$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx $$ From the fact that actually $$ δ(sin(x))=...

For a standard one-dimensional Brownian motion W(t), calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$I can't figure out how the middle term simplifies. $$ \mathsf E\left(\int_0^T W_t\mathrm dt\right)^2 = \mathsf E\left[T^2W_T^2\right] - 2T\mathsf E\left[W_T\int_0^T...

Hey, sorry for the cluncky title. It was rathet difficult to summarise what I'm talking about here. I want to know if it's possible to define ##f(x)## and ##g(x)## in such a way that ##∫f(x)g'(x)dx## has no indefinite solution while ##∫f'(x)g(x)dx## does have an indefinite solution. Any help...

I considered the work done by the frictional force in an infinitesimal angular displacement: $$dW = Frd\theta = (kr\omega) rd\theta = kr^{2} \frac{d\theta}{dt} d\theta$$I now tried to integrate this quantity from pi/2 to 0, however couldn't figure out how to do this$$W =...

Homework Statement: The question is in Attempt at a solution. Homework Equations: x=tanA/b I tried by substituting x=tanA/b but it did'nt helped.Now I cannot think of any other thing to do.Help.

Lets take for example Gauss's law in integral form. Suppose at time ##t## we have charge ##q(t)## (at the center of the gaussian sphere) enclosed by a gaussian sphere that has radius ##R>>c\Delta t##. At time ##t+\Delta t## the charge is ##q(t+\Delta t)## and if we apply gauss's law in integral...

I have to prove that, for a non-increasing function ##g(x)## the following inequality is true: $$k^2\int_k^\infty g(x) dx\leq\frac{4}{9}\int_0^{\infty}x^2g(x) dx$$ This exercise is from the book Mathematical methods of statistics by Harald Cramer, ex. 4 pg 256 Following the instructions of the...

In Physics/Electrostatics textbook, I am in a situation where we have to find the electric field at a point inside the volume charge distribution. In Cartesian coordinates, we can't do it the usual way because of the integrand singularity. So we use the three dimensional improper integral...

Summary: Does Richard Feynman's multiple histories ignore alternative histories? Did Richard Feynman's multiple histories (https://en.wikipedia.org/wiki/Multiple_histories) ignore the existence of other alternarive histories or paths? I ask this referring to this comment from this page...

AP Calculas Exam Problem$\textsf{Using $\displaystyle u=\sqrt{x}, \quad \int_1^4\dfrac{e^{\sqrt{x}}}{\sqrt{x}}\, dx$ is equal to which of the following}$ $$ (A)2\int_1^{16} e^u \, du\quad (B)2\int_1^{4} e^u \, du\quad (C) 2\int_1^{2} e^u \, du\quad (D) \dfrac{1}{2}\int_1^{2} e^u \, du\quad...

Dear Every one, I am having some difficulties with computing an element in the Integral dihedral group with order 6. Some background information for what is a group ring: A group ring defined as the following from Dummit and Foote: Fix a commutative ring $R$ with identity $1\ne0$ and let...

Show, that the identity \[\int_{0}^{1}\frac{x^{n-1}+x^{n-\frac{1}{2}}-2x^{2n-1}}{1-x}dx = 2\ln2\] - holds for any natural number $n$.

Sorry for the silly question! If we start of with the relationship $$\int_{x_{1}}^{x_{2}} F dx = KE_{2} - KE_{1}$$ and then state that at position x1 the velocity (and hence also kinetic energy) of the particle is 0, and at x2 its velocity is v, is it sloppy or valid to write the integral...

I need to solve this integral which I suppose is an elliptic integral but don't know what kind, I'm not that familiar with them. Mathematica says that it can be expressed with elementary functions and gives the solution: ## -\frac{2\...

Hi, I was trying to numerically integrate the following inverse Fourier transform integral,, using the code below. The plot is also shown below. The plot looks good which means the result is good as well. By the way, I was getting a warning which I quote below the code. % file name...

Find \[\lim_{n\rightarrow \infty}\int_{0}^{\infty}\frac{e^{-x}\cos x}{\frac{1}{n}+nx^2}dx.\]

Attempt at solution: Writing the chain rule for ## E(V,T) ##: ## dE = \frac{\partial E}{\partial T}dT + \frac{\partial E}{\partial V}dV ## Then, integrating the differential: ## \int{ dE } = \int{ \frac{\partial E}{\partial T}dT } + \int{ \frac{\partial E}{\partial V}dV } ## If I put the...

Using Lagrange multiplier ##\lambda## (only one is needed) the integral to minimize becomes $$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$ Using E-L equations: $$\frac {\partial F}{\partial x} - \frac d {d \tau}...

Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...

The integral has the form: $$\frac{s^2\nu^4}{(2\pi)^2}\int_{-1}^1 u(1-u^2)k_f^5[|r_1\chi_1|^2+|r_1\chi_2|^2-|r_1|^2\chi_1^*\chi_2\cos(2k_f\sqrt{u^2-\nu^2}a)-|r_1|^2\chi_2^*\chi_1cos(2k_f\sqrt{u^2-\nu^2}a)]\, du$$ ##r_1,\chi_1## and ##\chi_2## are also imaginary functions of u, because the form...

I would like to compute the triple integral of a function of three variables [FONT=Times New Roman]$f(x,y,z)$in R. I am using the package Cubature, Base, SimplicialCubature and the function adaptIntegrate(), Integrate and adaptIntegrateSimplex(). The integrand is equal to 1 only in certain...

the graph of x= √4-y^2 is a semicircle or radius 2 encompassing the right half of the xy plane (containing points (0,2); (2,0); (0-2)) the graph of x=y is a straight line of slope 1 The intersection of these two graphs is (√2,√2) y ranges from √2 to 2. Therefore, the area over which we...

Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...

I don't entirely get why we usually say that only the shortest path contributes in the path integral. If you calculate the volume of nth fresnel zones which is the locus where the path length is between n-1 and n wavelengths from the shortest path in 3 dimensions, they are the same I believe. So...

Problem Statement: Requesting for re check Relevant Equations: Requesting for re check In this eq.A4 putting ##v=Hr+u## the first integrand in eq.A5 is coming as ##H(r(\nabla•u)-(r•\nabla)u+2u)\ne\nabla×(r×u)## Am I right?? Can I request anyone to please recheck it... using this the author...

Dear Physics Forums people, My problem lies in understanding how the following line integral, which represents work done by the gravitational force, was calculated Specifically, in the integral after the 2nd = sign, they implicitly used \hat{r}\cdot d\vec{s} = dr I wish to understand what...

I have a quick question about the work done concept here, especially the line integral part of it. So I understand the fact that the work done from getting from point A to B is: \int_{a}^{b} \vec F \cdot d\vec r . However, within the context of electric fields, when we define electrostatic...