What is Integrating: Definition and 971 Discussions

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.

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  1. F

    How Can I Correctly Integrate e^(ix)cos(x)?

    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...
  2. I

    Integrating Implicit Functions

    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...
  3. W

    Calculating Volume: Integrating Pressure & Area

    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
  4. EastWindBreaks

    Integrating Factor: Homework Help

    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...
  5. M

    Integrating equations of motion with Earth perturbations

    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...
  6. G

    Integrating Factor Homework: Correctness & Complexity

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

    MHB Integrating $\sin^4 x$ to get $\cos^2 2x$

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

    MHB Integrating $\cos(u)\cos(\sin(u))$

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

    MHB Integrating Fouries series problem

    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...
  10. F

    Impossible Integration involving cosx/sqrt(a-bcosx)

    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...
  11. Rumplestiltskin

    Integrating for area of intersection

    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...
  12. T

    Integrating a theta function/2d probability density function

    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 $$...
  13. J

    Integrating Gaussian Distribution (QM)

    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...
  14. A

    Integrating force with respect to acceleration

    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.
  15. H

    Problem integrating a double integral

    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...
  16. C

    Integrating with a given substitution

    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 ⋅...
  17. J

    Integrating over a cross product?

    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
  18. Oribe Yasuna

    Integrating dx / (4+x^2)^2 using Trigonometric Substitution

    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 =...
  19. ecoo

    Integrating differential equations that have ln

    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...
  20. A

    Integrating a square root function

    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...
  21. A

    Integrating a polynomial with a square root

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

    MHB Integrating Factor: Solve x ln(x) dy/dx = xe^x

    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))
  23. Mr Davis 97

    Integrating the gravity formula?

    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?
  24. yango_17

    Manually integrating to find flux through hemisphere

    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...
  25. S

    Problem integrating over a sphere.

    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...
  26. J

    Choosing Integrating constants for Electric Field

    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...
  27. Priyadarshini

    Integrating (4x+1)^1/2 | Homework Help

    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
  28. A

    Solving (x+1)^2 dx for Integration - Step by Step Guide

    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...
  29. M

    Inconsistent values when integrating [Python]

    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...
  30. M

    Integrating Gaussian in polar coordinates problem

    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...
  31. vktsn0303

    What is an integrating factor exactly?

    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...
  32. neosoul

    Integrating Condensed Matter Physics and Materials Science

    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...
  33. J

    Integrating acceleration sin wave

    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...
  34. M

    Python: Integrating a function but getting large errors

    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 =...
  35. B

    Integrating parametric equations

    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...
  36. M

    MHB Integrating (triple) over spherical coordinates

    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
  37. P

    Finding an Integrating Factor for Solving Differential Equations

    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...
  38. B

    Physical insight into integrating a product of two functions

    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...
  39. Europio2

    Integrating 1/(1-x)^2 its making me crazy

    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...
  40. C

    Integrating law of gravity into a parabola?

    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...
  41. W

    Integrating in Polar Coordinates: Ω Region

    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...
  42. andyrk

    Integrating Net Force: Limits on LHS?

    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...
  43. C

    Physics w/Calculus II Electric Potential, Non-Uniform Charge

    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)...
  44. MexChemE

    Integrating factor vs. Laplace. Engineering problems

    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...
  45. D

    Problem integrating gamma ray absorption model

    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...
  46. TyroneTheDino

    Integrating Using a Substituation

    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...
  47. I

    Integrating an inverse square to find U

    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...
  48. Marco Lugo

    D.E. Linear equation with integrating factor

    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)...
  49. 2

    How can calculus be applied to determine wind force and moments on a wall?

    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...
  50. karush

    MHB Integrating a Product of Trig Functions

    $$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$ 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.
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