Integration Definition and 1000 Threads

Consider the partition function ##Z[J]## of the Klein-Gordon theory ##Z[J] =\int \mathcal{D}\phi\ e^{i\int d^{4}x\ [\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}+J\phi]} =\int \mathcal{D}\phi\ e^{-i\int d^{4}x\ [\frac{1}{2}\phi(\partial^{2}+m^{2})\phi]}\ e^{i\int d^{4}x\...

Homework Statement Hi, I'm doing a variation of parameters problem for my differential equations class. It requires solving the integral: ∫ex t-2 dt I am sure my professor did not give me an impossible integral and that there is some algebraic "trick" to solving it, but despite going through...

I would like to evaluate the following integral: ##\displaystyle{\int_{-\infty}^{\infty} dp^{0}\ \delta(p^{2}-m^{2})\ \theta(p^{0})}## ##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\ \delta[(p^{0})^{2}-\omega^{2}]\ \theta(p^{0})}## ##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\...

I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##. Using their notation, consider a contour ##\mathcal{C}## with...

Homework Statement http://imgur.com/a/qlQ5z Homework Equations i=i(o)+1/L integration(v0) dt formula is in the attemp at a solution. The Attempt at a Solution http://imgur.com/a/HVjl1 For the interval 2<t<infinite . I understand...

Homework Statement Homework Equations spherical Jacobean The Attempt at a Solution I have (sorry, have to capture my work, too hard to type) then the integration of p3 ep2 = 1/2 ep2 (p2-3/2) ??

Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...

I can't understand the solution to Problem 1.4(a). The solution is the following: What puzzles me is that ρ(θ)dθ=ρ(x)dx ? Why are they equal?

Is their a tutorial or a reference on how to decompose a function, specifically Fourier and Legendre decomposition, for numerical integration? The method I am going to use for the numerical integration is the Gauss Quadrature, and I suppose I need to decompose my function for the rule to work...

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...

Homework Statement in this problem , i couldn't express all the x any y in terms of u( refer to the circled part ) ... so , i have problem to proceed my subsequent steps ... Homework EquationsThe Attempt at a Solution i let u = (x^2) + (y^2) ... [/B]

$$\int(1-y^2)^\frac{1}{2}\,dy$$ I did trig substitution $$y=\sin\theta$$ $$dy=\cos\theta\,d\theta$$ $$\int(1+\cos2\theta)d\theta$$ $$\arcsin\,y+\frac{1}{2}\sin(2\arcsin\,y)+c$$ How do I get rid of the arcsins?

I need some help with this integran $$\int\frac{2x^2}{2x^2-1}dx$$I can't seem to solve this using the techniques that I know. What method should I use?

So I need to compare the results of the volume formula of a cylinder to the results of the integration. In geometry, you learn that the volume of a cylinder is given by V = πr2h, where r is the radius and h is the height of the cylinder. Use integration in cylindrical coordinates to confirm the...

For part of a proof of a differential equations equivalence, we needed to use that $$\int_0^t [\int_0^s g(\tau,\phi(\tau))\space d\tau]\space ds = \int_0^t [\int_\tau^t ds]\space g(\tau,\phi(\tau))\space d\tau$$ I understand that the order is being changed to integrate with respect to s first...

Homework Statement Find the value of the integral ## \int_0^\infty dx \frac{\sqrt{x}}{1+x^2} ## using calculus of residues! Homework EquationsThe Attempt at a Solution This is how I did it: ##\int_0^\infty dx \frac{\sqrt{x}}{1+x^2}=\frac 1 2 \int_{-\infty}^\infty dx \frac{\sqrt{|x|}}{1+x^2} ##...

Homework Statement Given the wave function of a particle \Psi(x,0) = \left(\frac{2b}{\pi}\right)^{1/4}e^{-bx^2} , what is the probability of finding the particle between 0 and \Delta x , where \Delta x can be assumed to be infinitesimal. Homework EquationsThe Attempt at a Solution I proceed...

Hi,I'd like to build a simulation (realtime) of space ships near a black hole (neutral, still or rotating possibly). Key features would be: 1) the ships are test particles that do not affect the metric a) possibly test rigid-bodies with GR consistent rotational DOF. 2) the ships can fire...

I used substitution but did not get a form I know how to integrate. What technique should I use here? $\int\frac{1}{x(x^2+1)}dx$ Thanks!

Can someone help with this problem, i tried to solve it using f'(x)/f(x) but couldn't figure it out.

Why this integral $\int\left\{\sqrt{{a}^{2}+{x}^{2}}\right\}dx$ needs integration by parts? Thanks Cbarker1

Hi, I am trying to analyze the an harmonic oscillator using kinematics. first i calculate the force applied by the spring (f = (-x)*k) then i calculate the acceleration (a = f/m) then speed (v= v0 + v0t + 0.5*a*t^2) and finally update x (x = x0+vt) this is a simplfied loop of my program...

Is it possible to find the volume of a sphere(i know the formula) using definite integration ? And if possible how to proceed ?? Thanks in advance

Cauchy integral theorem states that the contour integration of a complex harmonic function along a closed simply connected path=0. What if this simply connected path is drawn over a Riemann surface of function like ##f(z)=\sqrt z##. Will that be possible in the first place? and will the...

$\large {S6.7.1.13}$ $\tiny\text {natural log Integration}$ $$\displaystyle \int e^{\sqrt[3]{x}} \, dx = 3\left(x^\frac{2}{3} -2\sqrt[3]{x} +2\right){e}^\sqrt[3]{x}+C \\ u=x^{1/3} \therefore 3{x}^{\frac{2}{3}} du = dx $$ $\text{not sure if this is how to start to get to a 3 term answer}...

micromass submitted a new PF Insights post Omissions in Mathematics Education: Gauge Integration Continue reading the Original PF Insights Post.

Homework Statement Q=-1*K(T)*(H*W)*(dT/dx)+((I^2)(p)(dx)/(H*W)) K(T)=(197.29-.06333333(T+273)) H=0.01905 W=0.06604 I=700 p=10*10^-6 Q=some constant Please separate and differentiate to solve for Q using variables of T and x. Boundaries: T: Upper=T1 (constant) Lower=T0 (constant) x: Upper=L...

Homework Statement You have been employed but(sic) the Mathematics Football League (MFL) to design a football. Using the volume of revolution technique, your football design must have a capacity of 5L ± 100mL. You must present a statement considering the brief below. Just a quick side note, I...

Hello! I am currently in Calculus 2 and we are dealing with hydrostatic force. I get the integration that happens but I always seem to have trouble with solving the word problems. With this problem, I realize that I use integrate by making an infinite number of rectangles on the 2 triangles. I...

$\Large {S6-7.1.24}$ $$ \displaystyle I=\int_{0}^{\pi} {x}^{3}\cos\left({x}\right)\,dx=12-3{\pi}^{2} \\ \begin{align} u& = {{x}^{3}} & dv&=\cos\left({x}\right) \, dx \\ du&={3x^2} \ d{x}& v&={\sin\left({x}\right)} \end{align} \\ $$ $$ \text{IBP} \displaystyle =uv-\int v\ du \\...

Integration as antiderivative Question: Why isn't a distinction made between anti-total-derivatives and anti-partial-derivatives in common usage of integration? Consider the functions ##G_1(x, t)## and ##F(x, t)## such that ##F(x, t)=\frac{d}{dx}G_1(x, t)=\frac{\partial G}{\partial...

I have been given the problem ∫√(ex-3) and we must use the substitution u = √(ex-3) I can start it off with u = √(ex-3) and du = exdx/2u and what I've been trying is to complete the square and go towards 2 ∫ u2du/((u2+4) -1) But I am not getting towards the answer, either I am doing...

Homework Statement The integral I want to solve is $$ D(x) = \frac{-i}{8\pi^2}\int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r^2\sin\theta$$ which I've reduced to $$ D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}} $$ by integrating over ##\theta##. However, I...

Homework Statement Find the arclength of $$r(t) = <{t}^2/2,{t}^3/3>$$ from t=-1 to t=1 Homework Equations I have used this equation for arclength $$\int_{-1}^{1}|{r}'(t)|dt$$ The Attempt at a Solution After integrating (using u substitution) I have the solution...

micromass submitted a new PF Insights post Some Misconceptions on Indefinite Integrals Continue reading the Original PF Insights Post.

$\Large{§8.8.15} \\ \tiny\text {Leeward 206 Integration to Infinity}$ $$\displaystyle \int_{e^{2}}^{\infty} \frac{dx}{x\ln^p\left({x}\right)}\,dx \,, p>1$$ $\text{not sure how to deal with this} $ $\text{since there are two variables x and p} $ $\text{answer by maxima is:'} $...

Homework Statement D1=6cm D2=2cm ( i have to prove that 2nd moment of area (J) of a circulal plate abouts its polax axis(zz) is equal to piD^4/32 ) Homework EquationsThe Attempt at a Solution J=pi6^4/32 - pi2^4/32 = 125.66 J=r1^2x2pirdr J=Integral 2pir^3d2 = 2pi integral(high 3 , low 1)...

$$\Large{§8.8. 14} \\ \tiny\text {Leeward 206 Integration to Infinity}\\ \displaystyle I=\int_{2 }^{\infty} \frac{1}{x\ln\left({x}\right)}\,dx \\ \begin{align}\displaystyle u& = \ln\left({x}\right) & du&=\frac{1}{x} \ d{x} \end{align} \\ \displaystyle I=\int_{2}^{\infty}\frac{1}{u}...

Homework Statement can someone explain about the RHS of EIy' and EIy'' ? how to get the RHS of EIy" from RHS of EIy' ?? It's not integration of dx , am i right? Homework EquationsThe Attempt at a Solution if it's integration of dx, it should look like this , right?[/B] EIy' = 0.25P(x^2) -...

Homework Statement ##∫e^x^2 + 2e^x^2x^2 dx##[/B]Homework EquationsThe Attempt at a Solution i let## u= x^2, ⇒ du = 2x dx, ⇒∫e^x^2 dx = e^x^2/2x ## is this correct? by using integration by parts.... i am getting ##xe^x^2-e^x^2/2x##

Homework Statement ## d/dx (ye^∫pdx = Py+ y'e^∫pdx## now i know how they got ## y'e^∫pdx## . How do you differentiate ##ye^∫pdx## to get the first part i.e## Py ## presumably by product rule? Homework EquationsThe Attempt at a Solution [/B] d/dx of e^∫pdx is equal to P ...how?

Homework Statement [/B] ##∫ (1/sin 2x)dx##Homework EquationsThe Attempt at a Solution let ##u = sin 2x, ⇒ du= 2cos2x dx## or ##sin 2x= 2 sin x cos x##...[/B] or ##∫ (1/sin 2x)dx = ∫( csc 2x)dx##

$\tiny\text{LCC 206 8.8.11 Infinite Intervals of Integration}$ $$\displaystyle I=\int_{1}^{\infty} {x}^{-2} \,dx = 1$$ $$I=\left[\frac{1}{x}\right]_1^\infty=\left| 0-1 \right|=1$$ $\text{the only way apparently to get 1 is to use absolute value ?}$ $\tiny\text{from Surf the Nations math study...

The answer here shows the answer as being but my limited knowledge of integrals begs me to asks where did the z go?

Hello can you help solve this problem $$\int_{}^{}\frac{ds}{\sqrt{s^2-0.01}}$$ I tried using method of substitution but I still could not find a good cancellation. Please tell me what to do. Thanks!

|3^xlog3dxI don't even know where to start. I know that the formula is |u.dv = uv - |v.du u=3^x v=log3

Homework Statement Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 5. (Use disk method) $$ xy = 3, y = 1, y = 4, x = 5 $$ Homework Equations [/B] The formula using for disk method is of the form: $$ \pi \int...

I need some help with an Integration. Here's the equation I = ##\int_0^∞ \frac {x^{n+1}} {1 + x^n} ⋅ e^{-x^2/{2 \sigma^2}} ## I have tried to solve the equation by simplifying first like let, ## \frac x {\sqrt 2 \sigma} = v ##, so the ##x = v \sqrt 2 \sigma## then, ##dx = \sqrt 2 \sigma d...

As there is a repeated root, the partial fraction decomposition we should use is: $\displaystyle \begin{align*} \frac{A}{x - 1} + \frac{B}{\left( x - 1 \right) ^2 } + \frac{C}{x - 2} &\equiv \frac{x^2}{\left( x - 1 \right) ^2\,\left( x - 2 \right) } \\ \frac{A\,\left( x - 1 \right) \left( x - 2...

Hi, I've read some high school "derivations" of ##E=m\cdot c^2## that all considered single photons with momentum ##p=E/c## that are absorbed or emitted from some massive object, changing its mass. So they actually only showed the incremental $$\Delta E=\Delta m\cdot c^2 .$$ Most of those...