Integral Definition and 1000 Threads

I have the following integration - $$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx $$ To solve this I did the following - $$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$ Which gives me -...

Solving integrals by hand is difficult and prone to errors, and the techniques such as integration by parts, partial fraction decomposition, and trig substitutions only work for a small subset of integrals and I do not see the point of avoiding technology like Wolfram Mathematica for...

Hey! 😊 I want to calculate the integral $$\int\frac{1}{(x+4)(x^2-8x+19)}\, dx$$ I have done the following : $$\frac{1}{(x+4)(x^2-8x+19)}=\frac{1}{67}\frac{1}{x+4}+\frac{1}{67}\frac{12-x}{x^2-8x+19}$$ and so we get \begin{align*}\int\frac{1}{(x+4)(x^2-8x+19)}\, dx&=\frac{1}{67}\int...

Greetings here is my integral Compute the volume of the solid and here is the solution (that I don't agree with) So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric...

Greetings! I have the following integral and here is the solution of the book (which I understand perfectly) I have an altenative method I want to apply that does not seems to gives me the final resultMy method which doesn't give me the final results! where is my mistake? thank you!

Hello! The integral in equation (16), at the paper, is: ##I = r \int_{-\pi}^{\pi} e^{-2kr\phi} ~d\phi ## My integration is as the following : ## I = - \frac{1}{2 k} e^{-2kr\phi} ~|_{-\pi}^{\pi} + C ##, so ## I = - \frac{1}{2 k} ( e^{-2kr\pi} -e^{2kr\pi})+ C ## Now how to use the initial...

I know that gradient is multi-variable derivatives. But, here line integration (one dimensional integral) had canceled gradient. How?

Hi! Some help with this problem would be much appreciated. The overlap integral is defined as ##S = \int \phi_A (\mathbf{r}_A) \phi_B (\mathbf{r}_B) \,d\mathbf{r}##. For the two orbitals, I have that $$\phi_A = \frac{1}{\sqrt{\pi}} \Big( \frac{1}{a_0} \Big)^{3/2} e^{-r_A / a_0}$$ for the 1s...

I'm trying to see if I can calculate the peak draw weight of my bow based on the draw length and the velocity of the arrow and a known shape of a curve, but I'm not quite sure how to make such a function, if there even is such a way. This is the shape of the draw weight plotted against...

This problem comes from fluid dynamics where Kelvin circulation theorem states, that if density "rho" is a function of only pressure "p", then closed line integral of grad(p) / rho(p) equals zero. It seems so trivial, so that no one ever gives reason for this claim. When trying to solve it...

Hi, It's not a homework problem. I was just doing it and couldn't find a way to change the integral limit from "x" to "t". I should end up with kinetic energy formula, (1/2)mv^2. I've assumed that what I've done is correct. Thank you! Edit: "E" is work done.

hi guys I have a question about whether or not the Fermi-Dirac Integral has Been solved, because i found a formula on Wikipedia that relates the Fermi-Dirac integral with the polylogarithm function: $$F_{j}(x) = -Li_{j+1}(-e^{x})$$ and in some recent papers they claim that no analytical...

Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted? $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$

I saw it somewhere but I did't know exactly what it meant. Could someone explain it to me like I am 5? Does it mean we integrate with respect to x n times? $$\int_{\mathbb{R}^n}f\, \mathrm{d}^n x$$

I sketched this out. With the z=0 and y=0 boundaries, we are looking at ##z \geq 0## and ##y \geq 0## I believe ##0 \leq x \leq 5## because of the boundary of ##y=\sqrt{25-x^2}##. This is my region ##\int_0^5 \int_0^\sqrt{25-x^2} x \, dydx ## ## =\int_0^5 xy \vert_{0}^{\sqrt{25-x^2}} \, dx##...

hi guys I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by $$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$ i was thinking about taking substitutions in order to...

I've been trying for a very long time to show that the following integral: $$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2 f_{L}^0f_{R}^0(1+f_{H}^0). $$ can be reduced to one dimension: $$ I_D...

I have been studying scattering process in QFT, but i am stuck now because i can't understand how this integral was evaluated: $$\int dp\space \frac{1}{\sqrt{p^2+c²}}\frac{1}{\sqrt{p^2+k²}}\space p² \space d\Omega \space \delta(E_{cm}-E_{1}-E_{2})$$$ Where Ecm = c + k, E1 is the factor in the...

Hey everyone, I have been struggling to find the expected value and median of f(x) = 1/2e^-x/2, for x greater than 0. I am just wondering how I do so? Thank you.

Let f be a 2 variables function. 1) ##f(x,y)=g(x)+h(y)\Rightarrow df=g'(x)dx+h'(y)dy\Rightarrow\int df=g(x)+k(y)+h(y)+l(x)=f(x,y),\textrm{ if } k=l=0## 2) ##f(x,y)=xy\Rightarrow df=ydx+xdy\Rightarrow\int df=2xy+k(y)+l(x)\neq f(x,y)## Why in the second case the function cannot be recovered ?

I'm going to type out my LaTeX solution later on. But in the meantime, can anyone check my work? I know it's sloppy, disorganized, and skips far more steps than I care to count, but I'd very much appreciate it. I'm not getting the answer as given in the book. I think I failed this time because I...

Let $F = (P(x,y),Q(x,y))$ a field of vector class 1 in the ring $A={(x,y): 4<x²+y²<9}$ and $x,y$ reals. I am having trouble to understand why this alternative is wrong: If $ \partial P /\partial y = \partial Q /\partial x$ for every x,y inside A, so $\int_{C} Pdx + Qdy = 0$ for every...

Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$

Hey guys ! I just need a little help on a integral I was trying to solve using feyman's technique. This is the integral from 0 to 1 of (sin(ln(x))/ln(x) dx, which has been solved in one of the videos of bprp, but I'm trying to solve it using a different technique, and I end up with a different...

Summary:: Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine. [Mentor Note -- thread moved from the technical math forums, so no Homework template is shown] Hi, I'm trying to find the answer to the following integral over the rectangle...

Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##. Attempt : I argue for or against each statement by putting it down first in blue and my answer in red. ##g(x)## is always positive : The exact...

It is clear that ##1-x^2## is equal to zero in both boundaries ##1## and ##-1##. So for me is interesting to think like this \frac{d^m}{dx^m}(1-x^2)^m=\frac{d}{dx}(1-x^2)\frac{d}{dx}(1-x^2)\frac{d}{dx}(1-x^2)... and...

Hi When we find integrals of Bessel function we use recurrence relations. But this requires that we have the variable X raised to some power and multiplied with the function . But how about when we have Bessel function of first order and without multiplication? How should we integrate it ?

Hi I have a gamma integral in which it is not obvious how I can fix the limits of integration in order to match the standard form of gamma function.I just need someone to tell me how to fix them. I mean the integral number 6 in the picture. You can see my attempt in the PDF .

Hello! I have a function ##f(t)## such that ##\int_a^b{f(t)dt}=f_0##. Is there a way to calculate (or bring it to a simpler form) ##\int_a^b{f(a)e^{t}}dt##? Thank you!

Since ##h## and ##k## are constants: $$\frac{h}{k}\cdot \int \frac{1}{y(h-y)} \ dy$$ Now, rewriting the integrating function in terms of coefficients ##A## and ##B##: $$\frac{1}{y(h-y)}=\frac{A}{y}+\frac{B}{h-y}\rightarrow B=A=\frac{1}{h} \rightarrow$$ $$\frac{1}{h}\int \frac{1}{y}\ dy +...

Question \[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \] How can I derive this answer?

Just a quick question: Does anybody know if there is a closed-form solution to this rather simple-looking definite integral? ##F(\lambda) = \int_0^{\infty} \dfrac{e^{-x}}{1 + \lambda x} dx## If ##\lambda > 0##, it definitely converges. It has a limit of 1 as ##\lambda \rightarrow 0##. But it...

I am not sure how does the integral was did here. More preciselly, How to go from the first line to the second line? Shouldn't it be $$\frac{4 \pi}{(2 \pi)^3} \int _{0} ^{\infty} p^2 e^{ip*r}/(2 E_p)$$ ? (x-y is purelly spatial)

The goal is to evaluate the below integrals. Please note ##x\in \mathbb{R}^3## The issue is that I do not understand the meaning of the integration boundary ##||y-x||=t## and the meaning of the notation ##dS(y)##. Would someone be kind to explain these notations to me like I am five? are ##x##...

Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right. I also tried to compute the right-hand side of...

Background information Earlier they've shown that some double integrals can be simulated if it contains pdfs. Ex: $$\int \int cos(xy)e^{-x-y^2} dx dy$$ Can be solved by setting: Exponential distribution $$f(x) = e^{-x}, Exp(1)$$ Normal distribution $$f(y) = e^{-y^2}, N(0, 1/\sqrt 2)$$By knowing...

Solution attempt: we make the substitution ##\frac{s}{2}=u## and ##ds=2du## to compute...

Hi guys, I got to solve this integral in a recent test, and literally I had no idea of where to start. I thought about substituting ##tan(\frac{x}{2})=t## in order to apply trigonometry parametric equations, integrating by parts, substituting, but I always found out I was just running in a...

The answer calculates the integral with ##du## before ##dv## as shown below. However I decided to compute it in the opposite order with different bounds. Here is my work: According to the definitions, $$\begin{cases} u=x+y\\ v=2x-3y \end{cases}$$ First we need to convert the boundaries in xy...

I am studying interacting scalar fields (from Osborn) using the path integral approach. We define the functional integral \begin{equation*} Z[J] := \int d[\phi] e^{iS[\phi] + i\int d^d x J(x) \phi(x)} \tag{1} \end{equation*} The idea is to differentiate ##Z[J]## with respect to ##J## and end...

Encountered this integral and I believe it converges by studying it numerically but not sure and was wondering how might I show it converges or diverges? Surely there must be a way. $$ \text{P.V.}\int_0^{\infty}\int_0^{\infty} \frac{\text{sinc}^2(1+\phi)v e^{-v}}{(e^{v/2}-1)(\phi-v)}dvd\phi $$...

That's my attempt: $$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$ Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets: $$-\frac{1}{2tan^2x}+c$$ But there is something wrong... what?

I am working with the Dilogarithm function and am having problems showing the following and was wondering if someone could help: $$ \int_0^1\int_0^y\left(\frac{1}{x-1}\right)\left(\frac{1}{y}\right)dxdy=-\frac{\pi^2}{6} $$ This is what I have so far: Iterating the first level: $$ \begin{align*}...

##y=x^2ln x-x## ##\frac {dy}{dx}=2x ln x+x-1## ##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x## it follows that, ##\int [2x ln x +x]\,dx##=##x^2 ln x## ##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx## ##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##

(a) i sketched a quarter of a sphere centred at x=0 , y=2 , z=0 (b ) ∫ ∫ √ (4-x2 - (y-2)2) dx dy with limits 0 < x < 2 and 0 < y <4 (c ) i converted to spherical polars and took the integrand as 1/r2 . the volume element is r2sinθ drdθd∅ This leads to the triple integral of sinθ with...

I have some variables that are uncertain, these are w_m = u.ufloat(0.1430, 0.0011) z_rec = u.ufloat(1089.92, 0.25) theta_srec = u.ufloat(0.0104110, 0.0000031) r_srec = u.ufloat(144.43, 0.26) and some constant values c = 299792.458 # speed of light in [km/s] N_eff = 3.046 w_r = 2.469 *...

I have an integral that depends on two parameters ##a\pm\delta a## and ##b\pm \delta b##. I am doing this integral numerically and no python function can calculate the integral with uncertainties. So I have calculated the integral for each min, max values of a and b. As a result I have obtained...

Goodd day, I have a question regarding an exercice I have already posted Bvu was very nice and provided this darwing I already have the solution But y question is : can we use the disk method? because as you can see even though the intersection was at x=-1 the sphere goes deep into the...