Integrate Definition and 650 Threads

I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function: ρ(r)=ρ0*e^(-r/h) A double integral in polar coordinates should do, but im not sure about the solution I get.

Hello, can someone help me to solve the following differential equation analitically: $$\frac{2 y''}{y'} - \frac{y'}{y} = \frac{x'}{x}$$ where ##y = y(t)## and ##x = x(t)## br Santiago

The integral is this one: ##\int (\dot x)^2 \, dt,## With ##x=x(t). ## I don't know how to solve that integral and I haven't find nothing to read about on how to proceed with this kind of (implicit function?) integrals without having the initial function.

Here I list my problem in the attachment.

I got the answer for this question but I was wondering about stepsize when it comes to a problem like this? Is there a way to change the step size? Would my step size change in line 20, or would it change in line 3? I tried changing line 3 to be t=0:100:17.1 but then I get a error message, so...

still typing...checking latex

Can anyone explain to me why the second one is the right? (See the attachment)

##\iiint 3 dr d\rho d\phi## The volume of a sphere is ##4\pi /3 r^3## so naturally the answer is ##4 \pi R^3## But when I integrate I do: ##3 \iint r |_0^R d\rho d\phi## ##3R \int \rho |_0^{2\pi} d\phi## ##6R\pi * \phi |_0^\pi = 6R\pi^2## What am I doing wrong?

How do you you integrate expressions like these? 1. ##\frac{1}{\sqrt{x^3 + 3}}## 2. ##\frac{1}{\sqrt{x^4 + 3}}## 3. ##\frac{1}{x \sqrt{x^5 + 3}}## 4. ##\frac{1}{x^2 \sqrt{x^3 + 3}}## 5. ##\frac{1}{x^3 \sqrt{x^3 + 3}}## 6. ##\frac{1}{x^4 \sqrt{x^3 + 3}}## 7. ##\frac{1}{x^5 \sqrt{x^3 + 3}}##...

What is ##\int \frac{1}{x^2 + 3} \ dx##? This is my attempt: ##x = \sqrt{3} \tan \theta## --> ##dx = \sqrt{3} \sec^2 \theta \ d\theta## ##x^2 + 3 = (\sqrt{3} \tan \theta)^2 + 3## ##= 3 \tan^2 \theta + 3## ##= 3 (\tan^2 \theta + 1)## ##= 3 \sec^2\theta## ##\int \frac{1}{x^2 + 3} \ dx = \int...

integral of 3/(x(2-sqrt(x))) using u=sqrt(x)? But not using partial fractions

Hi PF! I followed someone's help on here and have the following code in python that performs monte carlo integration from math import * from random import * def integrate(alpha): # MONTE CARLO INTEGRATION OVER NON-RECTANGULAR DOMAINS def f(pt): # RETURN INTEGRAND AS FUNCTION...

So it's basically a half circle with radius a. y = asin(t) $$\int_0^{\pi} asin(t) dt = -acos(t) |_0^{\pi} = 2a$$ The book says the answer is ##2a^2##, but maybe that's wrong?

Mentor note: Moved from a technical forum section, so missing the HW template. Summary:: Integrate acceleration when a = f(v) when separation of variables is not trivial, ie a = k +v^2 When acceleration is a function of velocity, ie there is a friction force, you would separate the variables...

A newbie who knows basic math is helping a five year old do his kindergarten project. The boy has to integrate a function ##f(x,y)## over the boundary of the first quadrant denoted ##\partial \Omega## where ##\partial\Omega = \{ x=0, y\geq 0 \} ∪ \{ x\geq 0, y=0 \} ## How would I explain to...

How do you integrate ##\frac{1}{\sqrt{x^3 + x^2 + x + 1}} \, dx##? Please give me some hints and clues. Thank you

How do you integrate ##\frac{1}{\sqrt{x^2 + 3x + 2}} dx##? I had tried using ##u = x^2 + 3x + 2## and trigonometry substitution but failed. Please give me some clues and hints. Thank you mentor note: moved from a non-homework to here hence no template.

I've always been taught that the indefinite integral of ##\frac{1}{x}## is ##\ln(|x|)##. Extending this to definite integrals, particularly over limits involving negative values, should work just like any other integral: $$\int_{-1}^{1} \frac {1} {x} dx = \ln(|-1|) - \ln(|1|) = \ln(1) - \ln(1)...

I'm looking for a general analytical solution to a particular ODE that comes up in neuroscience a lot. My feeling is that such a solution can't be obtained, otherwise someone would have presented it by now, but I don't have a good understanding of why it is so hard to solve. The equations as...

Does anyone know if it is possible to develop a fully analytical solution for a leaky integrate and fire neuron driven by arbitrary time-varying current? Here's what I have so far (setting as many possible constants to 0 and 1): The equations: ## \dot{V} = - V + I(t) ## and if ##V(t) = 1##...

I'm pretty confused here because after getting stuck on this problem, I tossed it into an integral calculator and it said the answer was 2 Si(x) + cos(x)/x + C. In intro calc we definitely haven't learned the Si(x) function or even gotten to any of the Taylor polynomial stuff yet. I tried IBP...

Hi, I'm trying to integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}## and ## \int \frac{pdz \cdot \hat p}{(\sqrt{p^2 + z¸^2)^3}}## I get ## \frac{\hat z}{(\sqrt{p^2 + z¸^2)}}## and ## \frac{2z \cdot \hat p}{(\sqrt{p^2 + z¸^2)}}## But the correct answer should be ## \frac{z \hat...

\[ \int_{0}^{\inf} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\] or \[ \int_{0}^{constant} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\] maybe application Residue theorem integral ? because this problem same the kramers kronig relation?

Ok I tried to follow an example but kinda got lost in this maybe more steps the answer is from W|A

I fell upon such a wrting : $$du=tan(d\theta)$$ How to integrate this ? I didn't try numerically but I thought of expanding the tangeant in series but then should for example $$\int d\theta^2$$ be understood as a double integration ?

In chapter 3 of Vibrations and Waves by French, there is a description about the equations of motion of a mass-spring system. It was written as shown in the attached picture: Here, m is the mass on the spring, k is the spring constant, x is the extension of the spring, and t is time. My...

I am working with a polynomial and wish to integrate over one of it's branch surfaces with high precision. The function is: ## -z^2 + z^3 + w (-4 z + 3 z^2) + w^3 (-2 + 8 z + 4 z^2 - 4 z^3) + w^2 (-z^3 - 9 z^4) + w^4 (6 - 8 z^2 + 7 z^3 + 8 z^4)=0## So I first solve the associated...

Hello everyone! Today in my BC Calculus (calc 2) class, we spent time trying to find ways to integrate the same function (all methods we're fluent with, but are reviewing for the upcoming AP test)... (I apologize in advance that I do not know how to use LaTex, but am trying to learn) f(x) =...

Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion: Mg*sin{α} - 1.5m*x(double dot)=0 I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...

How do I setup an integral to integrate over the following equation: V(t) = 1/(R*C) integral to t Vin(t) dt This is the capacitor voltage over time formula. I want to integrate over a sine wave from 9 to 81 degrees. Frequency of 120Hz, amplitude of 120V. The formula I used in wolframalpha is...

Homework Statement [/B] Evaluate ##\int_0^∞ \frac{\tan^{-1}(\pi x) - \tan^{-1}x}{x}dx ## 2. Relevant information This problem comes after a chapter on “multiple integrals” and so, in this context, I realized we could rewrite the single integral as a double integral: $$ \int_0^∞ \int_1^{\pi}...

How do we integrate this function? It is possible if the range is from 0 to infinity, but from xg to infinity? This equation comes from page 512 of the 1961 paper by William Shockley and Hans J. Queisser.

Hello I am studying Landau Mechanics (3rd ed.) In chapter III Integration of the Equations of Motions §15. Kepler's problem page 36 M(angular momentum), m(mass), E(mechanical energy), and α are constant. How to integrate it? Please help me...

Hi PF! Attached are two plots of the same function, one uses a parameter ##a=89.9## and the other uses ##a=89+9/10##. As you can see both functions are very different. Also, using NIntegrate over each gives two different outputs respectively, ##-6.91846*10^{12},-6.91949*10^{19}##. My question...

Homework Statement I've got to integrate the following $$ \int dx =\int \frac {d\phi} {\phi \sqrt {1 - \phi²}}. $$ Homework Equations [/B] I already know the answer but not how to get it. The answer that I got from solution is ## x = \operatorname {arcsech}{\phi} ##. The Attempt at a...

##\int d^4 x \sqrt {g} ... ## if I am given an action like this , were the ##\sqrt{\pm g} ## , sign depending on the signature , is to keep the integral factor invariant, when finding an eom via variation of calculus, often one needs to integrate by parts. When you integrate by parts, with...

Hello, I just want to clarify some things with a simple exercise: I have the equation ## \frac{\partial^2 f}{\partial A^\mu \,\partial A^\nu} = 0## and I want to integrate it once assuming that ## f=f(A^1,A^2,...,A^n)=f(A^\rho) ##. I think the solution should be ## \frac{\partial f}{\partial...

1. The problem statement, all given data I've been working through one of the lessons for my HNC and I'm totally stuck on how they got from 27.84x10^-6 to 3.593x10^4. I can follow it all fine including the integration after that section it's just the inbetween that I can't seem to get my head...

I'd like to integrate a function over a closed circle-like contour around an arbitrary point on a torus and I assume I would use the expression: $$ \int_{t_1}^{t_2} f(x,y,z) \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$ And I cannot come up with an explicit parameterization of the variables in terms of...

I'm working on a guitar amp distortion emulation which is waveshaping based on the following equation: f(x)= x/|x| * tanh (c * |x|^b) This looks like this: So the idea is values of "x" (the raw guitar signal amplitudes) are fed in and get soft then hard limited to an output of y=+/-1. As...

Hi PF! I'm doing several operations involving integrating Legendre polynomials. Since I am trying to loop through, my approach is something like N = 5; a = 0.5; x =linspace(-1,1,100); for k=1:N v(:,k) = legendreP(k+1,x)-legendreP(k+1,a)/legendreP(1,a)*legendreP(1,x); end% for j where I...

Homework Statement Please see the attachment. This isn't a question, but more so my understanding of the book. Please read the middle paragraph. So I'm suppose to take an integral of cos(x)sin(x) Wolfram and my book give the answer as -(1/2)cos(x)^2, but when I did it I got it as...

I want to integrate this: \int_0^∞ re^{-\frac{1}{2σ^2} (r-iσ^2q)^2} \, dr. If I change the variable r into t with this relation:r-iσ^2q=t, then the integral becomes\int_{-iσ^2q}^∞ (t+iσ^2q)e^{-\frac{1}{2σ^2} t^2} \, dt so it seems I cannot use the famous gaussian integral formula. But I got the...

This is the continuation of the below thread: https://www.physicsforums.com/threads/what-is-integral-tan-2x-dx.856530/ Can someone please tell me how to integrate tan 2x dx?

Homework Statement Integral of 1/sqrt(1+x^2) dx Homework Equations sin^2theta`+cos^2theta=1 1+tan^2theta=sec^2theta The Attempt at a Solution I plugged x=tant --> dx=sec^2t dt => integral of 1/sqrt9(1+tan^2t) sec^2t dt = integral of t = tan^-1t + C However, another answer I've seen involves...

Homework Statement integrate 1/(x+x^3+2) Homework EquationsThe Attempt at a Solution I have tried to use partial fraction but the process is very complicated. Are there any faster methods? It is one of the ten questions in a 50 minutes elementary level test. Here is my attempt, ∫ 1/(x^3+x+2)...

Hello all, I was just wondering if there is any rules for integrating a function with respect to it's own derivative. That is to say ##\int _{ }^{ }f\left(x\right)d\left(f'\left(x\right)\right)## or ##\int _{ }^{ }yd\left(\frac{dy}{dx}\right)## Thank you in advance for your time :)

Hello, I was looking over my notes and I was trying to figure out when we integrate Q enclosed when Q = ρ*d(volume). If there's one thing I've learned from physics II you only integrate when a field is non-uniform. I'm just wondering how we know when it's uniform (usually the problem will tell...

Homework Statement This is not a homework but since asked me I'm posting it here. I know how to intergrate by parts and can do this using formula But I'd like to do this using the tabular method Question Integrate xcos(x^2) using tabular methodHomework Equations The table with derivative on...