Integrating Definition and 971 Threads

I'am trying to prove \int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix} Wolfram tells so http://integrals.wolfram.com/index.jsp?expr=e^%28i*x%29cos%28x%29&random=false But I am stuck in obtaining the first term: My step typically involved integration by parts: let u=e^{ix}cos(x) and...

In one of the homework sheets my teacher gave us, we had to calculate area geometrically (meaning no integration was used). Some parts, she said, we needed to just eyeball which I hate doing. In this case the top left portion of a circle described by the equation...

Homework Statement why the formula of volume is given by integral of P and dA , the integral of P and dA would yield Force , right ? Homework EquationsThe Attempt at a Solution

Homework Statement hello, I was reading through the textbook and I have a hard time to understand this part: Homework EquationsThe Attempt at a Solution haven't been dealing with derivatives for a while, i don't understand how it got ln |u(t)| from the first equation. Am I treating the...

Hello to everybody, i have been programming an n-bodies integrator in MATLAB, in an earth-centric ECI perturbations framework. The main objective is to 'write down' in a procedure an interesting part of phisics, and secondly (at the very end of it) to get a complete integrator for meteor orbits...

Homework Statement is my integral R(x) and R(y) correct ? why it look so complicated ? Homework EquationsThe Attempt at a Solution

$$\int \sin^4\left({x}\right) dx \implies\int \left(\sin^2 \left({x}\right)\right)^2 dx$$ $$\implies \frac{1}{4}\int\left(1-\cos\left({2x}\right)\right)^2 dx \implies \frac{1}{4}\int\left(1-2\cos\left({2x}\right)+\cos^2 \left({2x}\right)\right)dx $$ Got this far...hope ok

$$\int\cos\left({\pi t}\right)\cos\left({\sin\left({\pi t}\right)}\right) dt$$ Couldn't find a way to simplify this so $$u=\pi t$$ $$du=\pi \ dt$$ From there?

As the 2nd part of a question, we start with the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): $ \delta(x-a) = \frac{2}{L} \sum_{n=1}^{\infty} sin \frac{n \pi a}{L} sin \frac{n \pi x}{L} $ The questions goes on "By integrating both...

Ok, I have encountered this one a few hours ago: \int \frac{cos x}{\sqrt{a - b cos x}}dx I have tried all techniques I know, integration by part, half-angle formula, substitution, trigo approach, and so on, but it seems, I could not integrate this one. I am no noob in integration, but this one...

How does it work that you can subtract y2 from y1 and integrate the product within defined limits for the area of their intersection (within those limits)? Maybe that's not the right terminology - you arrive at the area for the region bounded by both functions. Is it just the same in practice...

Homework Statement A two-dimensional circular region of radius a has a gas of particles with uniform density all traveling at the same speed but with random directions. The wall of the chamber is suddenly taken away and the probability density of the gas cloud subsequently satisfies $$...

Homework Statement I am struggling with one of the end of chapter questions in my QM textbook (see attachment as I don't know how to show calculus on PF). It has thrown me because the chapter introduces some of the key principles in QM by talking about probability but then it randomly chucks in...

So stupid question...but I integrated force with respect to acceleration and got 1/2(ma^2). Is there any "meaning" to this equation. I thought it was jerk but dimensional analysis doesn't give me units of m/s^3 but instead m^2/s^4 which makes no sense.

Hi, could you please help with the integration of this equation: $$\int_{x}\int_{y}\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\,dydx$$ where ##u(x,y)## . From what I remember, you first perform the inner integral i.e. ##\int_{y}\frac{\partial}{\partial...

Homework Statement Using the substitution x = 2sinθ, show that \int \sqrt{4 - x^2} dx = Ax\sqrt{4 - x^2} + B ⋅ arcsin(\frac{x}{2}) + C whee A and B are constants whose values you are required to find. Homework EquationsThe Attempt at a Solution x = 2sinθ \frac{dx}{dθ} = 2cosθ dx = 2cosθ ⋅...

Lets look at the force on a wire segment in a uniform magnetic field F = I∫(dl×B) I am curious if, from this, we can say: F = I [ (∫dl) × B] since B is constant in magnitude and direction

Homework Statement Evaluate the integral: integral of dx / (4+x^2)^2 Homework Equations x = a tan x theta a^2 + x^2 = a^2 sec^2 theta The Attempt at a Solution x = 2 tan theta dx = 2sec^2 theta tan theta = x/2 integral of dx / (4+x^2)^2 = 1/8 integral (sec^2 theta / sec^4 theta) d theta =...

Hey guys, I have a question concerning the rewriting of a differential equation solution. In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to...

Problem statement: Find the arc length of the curve defined by x = √t and y = 3t -1 on the interval 0 < t < 1attempted solution: dx/dt = 1/2t-1/2 , dy/dt = 3 and dx = dt/ 2√t dy/dx = 6√t length = ∫01 √(1 + (6√t)2) .dt/ 2√t = ∫01 √1 + 36t) dt/2√t now I'm stuck with a product that is very...

1. Integrate the following: (4x - x^2)^1/2 dx 2. Any assistance would be appreciated.3. Honestly don't know where to start.

How do you solve x ln(x) dy/dx = xe^x using the integrating factor? So far, I have put it into standard form. dy/dx + y/(xln(x))=xe^x/(x(ln(x))

I decided to integrate the formula ##g = \frac{GM}{r^{2}}##, and I ended up getting ##v_{g} = -\frac{GM}{r}##. What is the meaning of the latter formula, and is it useful in anyway?

Homework Statement Basically, I am being asked to calculate the electric flux through the top of a hemisphere centered on the z-axis using "brute force integration" of the surface area. Homework Equations Gauss' Law The Attempt at a Solution Using intuition and Gauss' law, I know that the...

Homework Statement Compute \int_S \vec{F} \cdot d\vec{S} \vec{F} = (xz, yz, z^3/a) S: Sphere of radius a centered at the origin.Homework Equations x = a \sin(\theta) \cos(\varphi) y = a \sin(\theta) \sin(\varphi) z = a \cos(\theta) Phi : 0->2 pi, Theta : 0->pi/2 . The Attempt at a...

Homework Statement A nonconducting disk of radius R has a uniform positive surface charge density sigma. Find the Electric field at a point along the axis of the disk at a distance x from its center. Assume that x is positive Homework Equations E=kq/r The Attempt at a Solution I know I'm...

Homework Statement Integrate (4x+1)^1/2 Homework Equations Integration (ax+b)^n dx= (ax+b)^(n+1)/ a(n+1) The Attempt at a Solution (4x+1)^(3/2)/ 4(3/2) = (4x+1)^(3/2)/6 but the actual answer is 3(4x+1)^(3/2)/8

Hi, I literally just registered so I have no idea about forum rules, also I'm not good in english. 1. Homework Statement The equation is (x + 1)^2 dx. U = (x+1) DU = 1DX Homework EquationsThe Attempt at a Solution Here I get (U^3) over 3 times DX = (x^3 + 3x^2 + 3x + 1) over 3 times 1 I...

I have a 2D Gaussian: ## f(x,y) = e^{-[(x-x_o)^2 + (y-y_o)^2]/(2*{sigma}^2)}## which I converted into polar coordinates and got: ## g(r,θ) = e^{-[r^2 + r_o^2 - 2*r*r_o(cos(θ)cos(θ_o) + sin(θ)sin(θ_o))]/({2*{sigma}^2})} ## The proof for how this was done is in the attached file, and it would...

I have a 2D Gaussian: ## f(x,y) = e^{-[(x-x_o)^2 + (y-y_o)^2]/(2*{sigma}^2)}## which I converted into polar coordinates and got: ## g(r,θ) = e^{-[r^2 + r_o^2 - 2*r*r_o(cos(θ)cos(θ_o) + sin(θ)sin(θ_o))]/({2*{sigma}^2})} ## The proof for how this was done is in the attached file, and it would...

While solving non-homogenous linear ODEs we make use of the integrating factor to allow us to arrive at a solution of the unknown function. Same applies to non linear ODEs where the ODEs are converted to exact differentials. But what I don't understand is how and why would someone have come up...

Hi PF Family, I'm a rising junior majoring in physics. I plan to enroll and be accepted into graduate school(s) such as UIUC, Princeton, and MIT. I know that requires much work and hard work. However, my problem is in choosing the right program. I want to be able to integrate CMP and Materials...

This is a really basic calc/physics question.If acceleration is defined as Acc= Asin(w*t), and I integrate this to get velocity, I get Vel=(-A/w)*cos(w*t)+C. If the velocity at t=0 is 0, then C=A/w. If I then integrate the velocity to get the displacement, I get...

My code: import numpy as np import matplotlib.pylab as plt import math import random from scipy import integrate R1 = .001 R2 = 7 def G(r,theta): sigma = random.randint(4000., 7000.)/1000. # width of beam is 4 - 7mm r0 = random.randint(0, R1*1000.)/1000. #random centroid theta0 =...

Homework Statement Why does \int_a^b \, y \; dx become \int_\alpha^\beta \, g(t) f^\prime(t) \; dt if x = f(t) and y = g(t) and alpha <= t <= beta? Homework Equations Substitution rule?The Attempt at a Solution I'm not sure how y = y(x) in the integrand turns into g(t). Isn't y a...

Hi, Set up the triple integral in spherical coordinates to find the volume bounded by z = \sqrt{4-x^2-y^2}, z=\sqrt{1-x^2-y^2}, where x \ge 0 and y \ge 0. \int_0^{2\pi} \int_0^2 \int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} r\ dz\ dr\ d\theta

1. solve the problem first finding an integrating factor of susceptible form. y(x+y)dx+(xy+1)dy=0Homework Equations form: M(x,y)dx+N(x,y)dy=0 intigrating factor: eint(1/n(dm/dy-dndx)dx The Attempt at a Solution u(x)=eint(1/(xy+1)(y(x+y)d/dy-(xy+1)d/dx)dx this reduces to eint((x+y)/(xy+1))dx...

I was wondering what the physical insight is of integrating a product of two functions. When we do that for a Fourier transform, we decompose a function into its constituent frequencies, and that's because the exponential with an imaginary x in the transform can be seen as a weighting function...

Homework Statement I know its easy, but I'm making a mistake somewhere that is making me crazy. I want to solve \begin{equation} y = \int 1/(1-x)^2 \cdot dx \end{equation} I use de sixth formula in this PDF, but it does not work...

Homework Statement Hey all, I've been learning some incredibly INCREDIBLY basic calculus on my own, so please take it easy on my stupidity. So here's what I was wondering. In a 1 dimensional theoretical system, let the acceleration experienced by an object = A, with the signage +/- indicating...

Homework Statement ∫∫dydx Where the region Ω: 1/2≤x≤1 0 ≤ y ≤ sqrt(1-x^2) Homework EquationsThe Attempt at a Solution The question asked to solve the integral using polar coordinates. The problem I have is getting r in terms of θ. I solved the integral in rectangular ordinates using a trig...

We have: dF = (bdx)2gxρ. Now, x varies from h-l to h to calculate the entire force from x = h-l to x = h. So we put lower limit h-l and upper limit h while integrating RHS. But we don't put any limits on LHS and simply leave it as ∫dF. Shouldn't LHS also have some limits? If so, what would they...

Homework Statement Picture of problem is attached A rod of length L lies along the positive y-axis from 0<=y<=L, with L = 0.400 m. It has a positive nonuniform linear charge density (charge per unit length) λ = cy2, where c is a positive constant and has a numerical value of 8.00 x 10-7. (a)...

Hello PF! We were doing mass balances on mixing tanks in one of my ChemE courses, and in one of the problems we arrived at the following DE: \frac{dC_B}{d \theta} + 0.025C_B=0.0125 e^{-0.025 \theta} Where CB is the concetration of salt in the tank and θ is time. The professor made us solve the...

Homework Statement In this lab various thicknesses of a few materials are placed between a source of gamma radiation and a couple different detectors. It is reasonable to assume that some small change in the thickness of the shielding would produce a proportional change in the intensity of the...

Homework Statement Use the substitution ##u=\frac{\pi} {2}-x## evaluate the integral ##\int_0^\frac {\pi}{2} \frac {\sin x}{\cos x + \sin x}dx##. Homework Equations [/B] ##\cos (\frac {\pi}{2}-x)=\sin x## The Attempt at a Solution [/B]I start by plugging "u" into the equation making the...

Hello everyone, This is probably going to come off as a very silly question. However, I have not had calculus in several years. I was angry that my physics textbook did not have a derivation of Electric Potential Energy. So, I finally came across it, and I see that the integration of the...

Homework Statement https://webwork.utpa.edu/webwork2_files/tmp/equations/2d/02a7e6a06f5b2424758fa01cc965f71.png with https://webwork.utpa.edu/webwork2_files/tmp/equations/80/81c176aa8964438a63eb096513245f1.png Homework Equations [/B] Standard form: y' + p(x)y = f(x)...

I am trying to get more familiar with using calculus in unfamiliar situations, although I am stuck when thinking about moments. I am considering a wall that is depressed 0.7m into the ground and sticks out above ground by 2.0m (and has a width of w metres) and I am assuming that wind speed...

$$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$ the ans the TI gave me was $\frac{\sqrt{6}}{4}$ the derivative can by found by the product rule. but really expands the problem so not sure how the $\frac{d}{dx}$ played in this.