Integration Definition and 1000 Threads

Homework Statement The speed of a pendulum bob moving in simple harmonic motion is given by v = 1.26sin(2πt) where v is in m/s and t is time in seconds. Homework Equations s = ∫ v dt The Attempt at a Solution v = 1.26sin(2πt) Integrating v yields s = -0.2cos(2πt) + c and solving for c...

Homework Statement [/B] Homework Equations ∫ f(x) g'(x) dx = f(x) g(x) - ∫ f '(x) g(x) dx f(x)=√(1+x^2) f '(x)=x * 1/√(1+x^2) g'(x)=1 g(x)=x The Attempt at a Solution ∫ √(1+x^2) * 1 dx =x * √(1+x^2) - ∫ x^2 * 1/√(1+x^2) dx Further integration just makes the result look further from what...

Homework Statement dz/dx=(3x2+x)(2x^3+x^2)^2[/B]Homework Equations ∫(3x^2+x)(2x^3+x^2)^2 dx The Attempt at a Solution I tried substituting (2x^3+x^2) Let t= 2x^3 + x^2 dt=6x^2+2x dx dt/dx= 6x^2+2x I can only solve till this point . I don't have any clue how to solve it further But how do we...

Can somebody help me? I am studying Faddeev-Popov trick, following the Peskin and Schroeder's QFT book, but I can't understand one thing. After they inserted the Faddeev-Popov identity, $$I = \int {{\cal D}\alpha \left( x \right)\delta \left( {G\left( {{A^\alpha }} \right)} \right)\det \left(...

v(x(t)), where v represents velocity and is a function of position which is a function of time. I have the equation: v dv/dx = 20x + 5 and want to solve for velocity. The way our professor solved it was by multiplying both sides by dx and integrating => ∫v dv = ∫20x+5 dx. I know doing this is...

This is I think a really dumb question, but I never got it, why do we have that dot symbol when we integrate. Like in gauss's law, we have ∫E⋅dA . why is that ⋅ there? Thank you for your help

In chapter 7 of "Modern Quantum Mechanics" 2nd edition by Sakurai and Napolitano, a treatment of the degenerate electron gas is given as an example for 2nd quantization. This treatments is mostly taken from the same material in chapter 1 of "Quantum theory of many particle systems" by Fetter and...

I'm having a tough time with this integral: $$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$ where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck. What contour should I...

I used trig substitution and got sqrt(x^2-9)+3*arcsin(3/x) which seems to be incorrect when I check it in my calculator and the textbook. I made a right triangle where one of the legs was sqrt(x^2-9) and it so happens that if you switch the leg the answer becomes sqrt(x^2-9) -...

Suppose we are solving a diffusion equation. ##\frac{\partial}{\partial t} T = k\frac{\partial^2}{\partial x^2} T## On the domain ##0 < x < L## Subject to the conditions ##T(x,0) = f(x) ## and ##T = 0 ## at the end points. My question is: Suppose we solve this with some integration scheme...

I was trying to derive the following results from 4B.8 as suggested by using the vector triple product identity but have been unsuccessful in deriving ##\vec{L_R}## and ##\vec{S_R}## in the end. After using the identity and finding the integrand to be ## \vec{E}(\vec{r}\cdot\vec{B}) - \vec{B}...

I'm trying to numerically solve the time dependent Schrödinger equation and I've been told that the best approach is to numerically integrate using a finite difference method, however I don't understand why I couldn't just use ode45 to solve it?! Is the finite difference (interpolation) method...

Homework Statement The question is pretty long and wordy so apologies in advance! Inside the Earth the gravitational field falls off linearly as one approaches the centre. An accurate description of motion in a very deep hole would therefore be to use Newton’s law, f = ma, but with the force...

Homework Statement Use spherical coordinates to find the volume of the solid enclosed between the spheres $$x^2+y^2+z^2=4$$ and $$x^2+y^2+z^2=4z$$ Homework Equations $$z=\rho cos\phi$$ $$\rho^2=x^2+y^2+z^2$$ $$dxdydz = \rho^2sin\phi d\rho d\phi d\theta$$ The Attempt at a Solution The first...

Hi all, i need help solving the following integral using Matlab: * tetam is a parameter and the integration is by alpha. the answer should be function of tetam * K33, K11 are constantsThanks , Chen

Homework Statement ∫ from 2 to 6 of dx/(x-4) Homework Equations None The Attempt at a Solution u=x-4 du=dx ln abs value x-4 abs value ln2-ln2=0 What am I doing wrong?

Homework Statement Hi everybody! I'm doing a problem about oscillations, and I must admit that a few things are still unclear to me about that subject. Can someone maybe help me? a) A onedimensional masspoint m is oscillating under the influence of the force F(x) = -c⋅x (c > 0). What is the...

Homework Statement Determine, by explicit parameterisation, ∫C dz f(z), where f(z) = z^n , n ∈ Z, n > 0 and C is the line segment from z = 0 to z = 2 together with the line segment from z = 2 to z = 2 + i. Homework Equations ∫ϒ f(z) dz = ∫ab dt f(ϒ(t))ϒ'(t) where ϒ'(t)≈dz/dt The Attempt...

Homework Statement Switch the order of integration: ∫∫f(x,y) dydx, with 0≤x≤π/2, 0≤y≤sin(x) The Attempt at a Solution ∫∫f(x,y) dxdy, with sin(x)≤x≤π/2, 0≤y≤sin(x) The upper bound of x is π/2, while the lower bound is sin(x). It's just the first 1/4 of the area of a sine wave. The upper...

Homework Statement Evaluate ∫∫D sin(x) / x dA, where A is the triangle with vertices (0, 0),(0, 1),(1, 1). Homework EquationsThe Attempt at a Solution I've set up the integral, with x≤y≤x^2, 0≤x≤1 for the upper and lower bands of each integral. dxdy I'm not sure how to go about it though. I...

I can't upload the notebook I guess, that would be a nice feature mods! (.nb extension) But here is my code , there seems to be a problem with integrating separately and then combining. $Assumptions = 0 < r < 1 && 0 < z < 1; FUNCTION = Expand[-(Pi*(-1 + r)*(-12*z + r^2*z + r*(-12 +...

In Matlab I am trying to use the composite Simpson's rule to find ##x_l## so that $$170=\int^{x_l}_0 \sqrt{1+(y')^2} dx = \int^{x_l}_0 \sqrt{1+\left( \frac{x^2}{68000} \right)^2} dx $$ For convenience this can be written as $$I(x) = 170 - \int^x_0 \sqrt{1 + (\frac{x^2}{68000})} dx$$ The...

After following the logical steps to derive something, I reached to the following integral: \int_0^{\infty}\frac{1}{x^2}\exp\left(\frac{A}{x}-x\right)E_1\left(B+\frac{A}{x}\right)\,dx where ##E_1(.)## is the exponential integral function, and ##A## and ##B## are non-zero positive constants. I...

I have to do a integration which goes like this: (V-M)(dP/dx)+3P(dV/dx)=0, (where M,P and V are constants). If you integrate with dx, you will get: ∫[(V-M)dP]+∫[3PdV]=0. which ultimately results in the answer M=4V. Now, i can put the first equation in this form also...

Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...

Homework Statement State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals. \begin{eqnarray*} (i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\ (ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...

Homework Statement If f: [0,1] \rightarrow \mathbb{R} is continuous, show that (n+1) \int_0^1 x^n f(x) \mathrm{d}x is in the range of f Homework Equations (n+1) \int_0^1 x^n f(x) \mathrm{d}x=\int_0^1 (x^{n+1})' f(x) \mathrm{d}x The Attempt at a Solution I tried integration by parts, but that...

Homework Statement \int_0^5 \int_0^2 \int_0^{4-y^2}\ \, dxdydx Change order to dydxdz Homework EquationsThe Attempt at a Solution I'm confused mainly because the limits are mostly numbers, not functions. I graphed the limits in @D and #d and this is what I got: \int_0^{4-y^2} \int_0^5...

I'm trying to integrate the Schrodinger equation ##i\hbar \frac{d}{dt} |\psi(t)\rangle = H |\psi(t)\rangle## with the initial condition ##|\psi(t_{0})\rangle=|\psi_{0}\rangle## to show that ##|\psi(t)\rangle = \exp(\frac{t-t_{0}}{i\hbar}H)|\psi_{0}\rangle##. I know how to plug in the solution...

When using the double integration method to solve for deflection, do you simplify the integral before integrating or do you just integrate it? When i tried it with simplifying and without simplifying, I get a different value for the constant which results in different answers. For example in...

I'm trying to solve numerically this multiple integral. But i don't know how to calculate it with Mathamtica or Sage software. $$\int{e^{-(\vec{v}^2_1+\vec{v}^2_2)}e^{-E(\tau)}}d\vec{r}_1d\vec{r}_2d\vec{v}_1d\vec{v}_2$$ $$E(\tau)=\frac{k}{\tau^2}e^{-\tau/\tau_0}$$...

Homework Statement dv/dt = -βv Integrate to find velocity as a function of time, assume the particle's initial velocity is v0. β is constant v = velocity (not constant) t = time Homework EquationsThe Attempt at a Solution dv = -βvdt ∫dv = -β∫vdt --------> limits of integration for the right...

Homework Statement Hi,I saw a statement in my physics notes like this(Anyway it is a maths problem): where L is a general differential operator.G is a green's function(I guess it is irrelevant) My question is related to the red line: Suppose we have this: ∂/∂x ∫ f(x-y)g(y) ∂y is it...

Homework Statement \frac{dy}{dx}=y^2-1 y(0)=3 Homework Equations \frac{dy}{dx}=f(y) \leftrightarrow \frac{dx}{dy}=\frac{1}{f(y)} The Attempt at a Solution \frac{dx}{dy}=\frac{1}{y^2-1} dx=\frac{dy}{y^2-1} \int dx=\int \frac{dy}{y^2-1}+C x=\int \frac{dy}{y^2-1}+C How do I integrate \int...

Homework Statement Solve the differential equation, dt/dv= 1/g (1/1-a^2*v^2) where a = (k/mg)^1/2 to yield v= 1/a(1-e^-2agt/1+e^-2agt). Homework Equations F=ma Newton's 2nd Law Integration Laws The Attempt at a Solution See image. I think I'm getting messed up on the integration laws.

Homework Statement I have the attached problem ,and Show my answer and solution, My answer is 3.14*(15 -8ln4) While book answer 3.14(15+8ln) Can anyone point to the mistake I made?

Homework Statement Find the integral \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm dx 2. The attempt at a solution I can't find a useful substitution to solve this integral. I tried x-2=\frac{1}{u},x=\frac{1}{u}+2,dx=-\frac{1}{u^2}du that gives \int \frac{1}{(x-2)^3\sqrt{3x^2-8x+5}}\mathrm...

Homework Statement Find the integral \int \frac{2\sqrt[5]{2x-3}-1}{(2x-3)\sqrt[5]{2x-3}+\sqrt[5]{2x-3}}\mathrm dx 2. The attempt at a solution...

Homework Statement Show the steps needed to obtain the equation for average molecular speed, cavg=√8RT/πM from the integral (from negative infinity to infinity) ∫v*f(v)dv where f(v) is the Maxwell distribution of speeds function f(v)=4π*(M/2πRT)1.5v2e-Mv2/2RT M is the molar mass of the...

I am attempting to normalize a wave function and need to integrate ##\int A^2*e^{(\lambda^2x^2)} dx## going from -inf to +inf. I tried to integrate this on Wolfram Alpha and this was the result. Upon integrating with the parameters the solution is as such. How does the erfi get removed? Do I...

1. The problem is as follows: ∫(√1+x^2)dx/(x) 2. Using trig sub --> x = atanΘ with a = √1 = 1. So x = tanΘ and dx = sec^2ΘdΘ. 3. Picture included of attempted solution. I tried u substitution with both u = secΘ and u=tanΘ but didn't have the right du...

We were taught of several integration technique, only to find one of those techniques came up as years of solving of our professor.. Can someone explain to me how substitution x = 1 / z dx = -dz / z^2 works for some problems? He called this reciprocal substitution, as what you can literally...

Is it possible to get Maple to show me step by step how to solve a complex contour integral? f := (x,y,z,v) -> (x+I*x*cos(v)+I*y*sin(v))^(-2) int(f(x,y,z,v),v=0..2*Pi) assuming(x,real,y,real,z,real,v,real) But I would like to know how Maple solves this step by step. I tried using the tutor...

I want to evaluate $$\int_{a+b-c}^s\,\text{d}x\, \frac{(-x+ab/c)^{\epsilon}}{(x+c-a-b)^{\epsilon+1} (a-x)},$$ where ##a,b,c,\epsilon## are numbers, and to be treated as constants in the integration. I put this into mathematica and an hour later it is still attempting to evaluate it so I aborted...

https://anhngq.wordpress.com/2013/04/25/the-cauchy-formula-for-repeated-integration/ Could someone please explain, bit by bit, how the formula works? For instance, why are we integrating with respect to sigma, up to sigma in the equation before it?

What does it mean by "making the time integration redundant" (5th line)? If I let ##t_2=t_1##, I will only get ##0=0-0##. Source: http://www.phys.ufl.edu/~maslov/classmech/flannery.pdf

Hey guys! I'm working on a problem for which i somehow just can't figure out what I'm doing wrong. This is the problem: I think I've figured it out, but somehow i think i make an error with the maths. First of let's determine the equation for the temperature (in Kelvin): $$T(z) =...

Homework Statement I need to integrate ##\frac{A}{2a\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac{e^{ik(x+x')}}{(b^2+k^2)}dk## I have tried substitution and integration by parts and that hasn't worked. I can see that part of it is the delta function, but I don't really know how to use that fact...

Homework Statement The problem is the integral attached Homework Equations sec2(u)=(1+tan2(x)) a2+b2=c2 ∫cos(u)=-sin(u)+C The Attempt at a Solution The solution is attached. I am wondering if someone could give me a hint where I went drastically wrong or where I possibly dropped a negative...

Hi, In transmission coefficient T= exp(-2sqrt(2m(U-E)/hbar^2)L), L, as I interpret it, is the distance of the potential barrier. I am wondering if I have N particles all with kinetic energy E, approaching the barrier, can I integrate the transmission coefficient over a distance from infinity to...