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. C

    Position vector for anti-clockwise circular motion derivation

    To derive ##\vec r (t)=(−Rsin(ωt),Rcos(ωt)) ## I start by integrating ##ω=\frac{dθ}{dt}## to get ##θ_f=θ_i+ωt##. Therefore since ##Δθ=θ## by definition since the angular displacement is always taken with respect to some initial reference line, then ##θ_f−θ_i=θ## , thus, ##\theta = \omega t##...
  2. C

    I Negative area above x-axis from integrating x^2?

    Suppose the following integration, ##\int_3^{-1} x^2 \, dx = \frac{1}{3}(-1)^3 - \frac{1}{3}(3)^3 = -\frac{28}{3}## However, if we have a look at the graph, The area between ##x = 3## and ##x = -1## is above the x-axis so should be positive. Dose anybody please know why the I am getting...
  3. C

    Integrating electric field of rod

    For this problem, However, I am trying to solve this problem using an alternative method compared with the solutions. My method is: ##\vec E = k_e \int \frac {dq} {r^2} \, dx ## ##\hat r## ##\vec E = k_e \int \frac {\lambda} {x^2 + d^2} \, dx## ## \hat r## If I let ## \hat r = \frac {-x\hat i...
  4. C

    Integrating Biot-Savart Law at a point

    Why when you integrate the Biot-Savart Law do we not include limits of integration on the left-hand side of the equation (for the differential magnetic field)? Would the lower limit be 0 and the upper limit be B? How would you tell? Many thanks!
  5. G

    Conquering the Integral of (1/x)*exp(-ax^2): A Scientific Inquiry

    Hopeless. I tried to use Taylor expansion but the zeroes and infinities go out of control really quick. I tried WolframAlpha and it gave a special function. What integrating trick am I missing? Or is it nonsense to solve it simply by hand?
  6. Buzz Bloom

    I I have an error in integrating to calculate the age of the Universe

    NOTE: I am attempting to convey the equations in this post into LaTerX format in Post #19. My result is way off. It is about 7.44 x 10^9 years. The values I use are: 1/H_0 = 14.4 X 10^9 years, M = Ω_m = 0.3103, and L = Ω_Λ = 1 - Ω_m = 0.6897. The equation I start with is the following. dt =...
  7. A

    Integrating motion equation to derive displacement

    ##\frac{dx}{dt} = \frac{dx_i}{dt} + \frac{d^2x}{dt^2}t## Multiplying dt on both sides and integrating we have ##\int_{x_f}^{x_i} dx = \int_{0}^{v_i t} dx_i + \int_{0}^{at} dvt## so ##x_f - x_i = v_it + at^2##, which is not right Where did I go wrong? I understand that if we substitute a for...
  8. Buzz Bloom

    I An odd result from integrating to calculate the age of the Universe

    The integration tool I am using is https://www.symbolab.com/solver/definite-integral-calculator . The following are the values of the five variables in the Friedmann equation with references of sources. I have also defined single letter variables I used for convenience...
  9. G

    How can I Easily Integrate F = ma into My Solution?

    F = ma -bv = ma -bv/m = a -bv/m = dv/dt dt = -mdv/bv ∫dt = -m/b ∫dv/v t = -m/b ln v -(b/m)t = ln v e^-(b/m)t = v
  10. L

    I Problem with integrating the differential equation more than once

    Starting from equation \frac{dy}{dx}=\int^x_0 \varphi(t)dt we can write dy=dx\int^x_0 \varphi(t)dt Now I can integrate it \int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^x_0\varphi(t)dt Is this correct? Or I should write it as \int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^{x'}_0\varphi(t)dt Best wishes in new year...
  11. Mayhem

    Integrating Logarithmic Functions with Binomial Terms

    Given the integral $$\int \ln{(e^x+1)} dx$$ we can rewrite this as the integral of the Taylor expansion of ##\ln{(e^x+1)}##. $$\int \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(e^x+1)^n}{n} dx$$ Which can then be rewritten using the binomial theorem: $$\int \sum_{n=1}^{\infty} \left [...
  12. A

    A Understanding Integrating Lambda with l1(y) and l0(y)

    I have the following function l1(y)=c1. Integrating lambda(t) = l1(y + a1t) from 0 to t gives (I/a1) (L1(y + a1t) - L1(y)), where L'1(x) = l1(x). Now I don't understand why that is. Similarly, I have the following function l0(y)=c0/y. Integrating lambda(t) = l0(y - a0t) from 0 to t gives -...
  13. P

    Integrating from R to r: Potential Contribution & Reasoning

    The potential contribution from R > 0 is simple. My next step is to integrate from R to r. With regards to the integration from R to r, the 2nd method gives a potential contribution that is the negative of the 1st method. What is the reason?
  14. Eclair_de_XII

    Integrating vector-valued functions along curves

    The following parametrizations assume a counter-clockwise orientation for the unit square; the bounds are ##0\leq t\leq 1##. Hypotenuse ##(C_1)## %%% ##r(t)=(1-t,1-t)## ##dr=(-1,-1)\,dt## ##f(r(t))=f(1-t,1-t)=(a(1-t)^2,b(1-t)^2)## ##f\cdot dr=-(a+b)(1-t^2)\,dt## \begin{align} \int_{C_1} f\cdot...
  15. karush

    MHB B.2.1.4 trig w/ integrating factor

    $\begin{array}{rl} \textit{Find } \mu(x): &\mu(x) =\exp\left(\int \dfrac{1}{x}\,dx\right)=e^{\ln{x}}=x\\ \textit{multiply thru by x} &xy^\prime+y=3x\cos 2x\\ \textit{rewrite as } &(xy)'=3x\cos 2x \\ \textit{}integrate &xy=\int 3x\cos 2x \...
  16. greg_rack

    Integrating Function by Parts: Help Needed!

    Hi guys, I've attempted to integrate this function by parts, which seemed to be the most appropriate method... but apparently, I'm getting something wrong since the result doesn't match the right one. Everything looks good to me, but there must be something silly missing :) My attempt:
  17. A

    Problem integrating with the disk method

    Good day I have the following exercice and it's solved using spherical coordinates I totally agree with the solution but I have issue to find out why mine does not work I used the the integration by disk I divided the region of integration to 2 A1 and A2 (A2 is the upper half sphere and A1 is...
  18. cwill53

    Integrating Mass of a Hollow Sphere: Multivariable Calculus Explained

    I know some multivariable calculus, I just want someone to walk me through the integration deriving the mass element dM and the integration of thin rings composing the hollow sphere. It would also be nice if you could show me doing it one way using the solid angle and one way without using the...
  19. E

    MHB Integrating a piecewise function?

    I have to find: g(1)= and g(5)= I have drawn the graph and I am a little unsure where to go from there. I know area is involved somehow but not entirely sure what to do. Any help is appreciated
  20. B

    Integrating with a Denominator of (1+x^2): A Step-by-Step Guide

    I think in the case of "n da" you can see the denominator (1+x^2) as a constant, so ∫ ( sin(a) + M^2 ) / ( 1 + x^2 ) da = ( 1 / ( 1 + x^2 ) ) * ∫ (sin(a) + M^2 ) da = ( 1 / ( 1 + x^2 ) ) * ( -cos(a) + (M^2)a ) = ( - cos(a) + (M^2)a ) / ( 1 + x^2 ) --- Is this the way to go? This is my...
  21. redtree

    I Integrating with the Dirac delta distribution

    Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...
  22. JD_PM

    I Assuming boundary conditions when integrating by parts

    Let's present two examples $$-\frac 1 2 \int d^3x'\big (-i \phi(x', t)\nabla^2\delta^3(x-x') \big )$$ Explicit evaluation of this integral yields $$-\frac 1 2 \int d^3x'\big (-i \phi( \vec x', t)\nabla'^2\delta^3(\vec x-\vec x') \big ) =\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec...
  23. Mayhem

    I Separation of Variables and Integrating over an Interval

    I was solving the van't Hoff equation over an interval ##[T_1 , T_2]##: The van't Hoff equation ## \frac{\mathrm{d} \ln K}{\mathrm{d} T} = \frac{\Delta_r H^{\circ}}{RT^2} ## which can be solved with separation of variables: ## d \ln K = \frac{\Delta_rH^\circ}{RT^2}dT ## ##\Updownarrow##...
  24. M

    Integrating ##\sigma=\chi\int{dA/A}## for a sphere

    I am trying to integrate ##\sigma=\chi\int\frac{dA}{A}## for a sphere. The answer is supposed to be ##\sigma(R)=\chi(R^2/R_0^2-1)##. The answer I keep getting is ##\sigma(R)=2\chi ln\frac{R}{R_0}##. I also tried doing it in spherical coordinates, and all I get for the integration of...
  25. Mayhem

    I Integrating x^ne^xn: Analyzing & Solving

    I was bored and tried to integrate ## x^n e^{xn} ##. I seem to be on the right track, but ultimately it is not entirely correct. Here is my work: Given is the integral $$I = \int x^ne^{nx}dx$$ where ##n \geq 1## We substitute ##t = nx## which gives us ## \frac{dt}{dx} = n \Rightarrow dx =...
  26. B

    Integrating Factor: Need Help Solving Excersice?

    I'm not sure if that is the right way to solve this excersice. Can someone maybe help and tell me if this calculation proofs the statement ?
  27. tworitdash

    A Integrating a function of which poles appear on the branch cut

    I have a complicated function to integrate from -\infty to \infty . I = \int_{-\infty}^{\infty}\frac{(2k^2 - \Omega^2)(I_0^2(\Omega) + I_2(\Omega)^2) - \Omega^2 I_0(\Omega) I_2(\Omega)}{\sqrt{k^2 - \Omega^2}} \Omega d\Omega Where I0I0 and I2I2 are functions containing Hankel functions as...
  28. T

    Integrating FE and Dynamics

    (CAVEAT: I am on the verge of retiring. And as I look back on my life, I realize how much I do not know. And I am using the convenience of time, now, to go back and ask the questions I always wanted to understand. So, please forgive me for these questions. They have always been my mind and I...
  29. S

    I Assigning a value for integrating a divergent oscillatory function to infinity

    There are meaningful ways to assign values to things like 1 - 1 + 1 + ... or 1 - 2 + 3 - 4 + ... In a similar spirit, is it possible to assign a value to the integral of a function like this: ##f(x)=x*sin(x)## or this one: ##g(x)=Re(x^{1+5i})## (Integrals from some value, say zero, up...
  30. E

    Integrating by Partial Fractions

    I was doing this problem from Griffith's electrodynamics book and can't figure out how to do this integral. The author suggested partial fractions but the denominator has a fractional exponent which I have never seen for partial fractions, and so, I am unsure how to proceed. The integral I am...
  31. N

    Integrating (sin(x))/x dx -- The limits are a=0 and b=infinitity

    U= 1/x dV= sin(x) dU = -1/x^2dx V= -cos(x). lim b--> infiniti (integral from [0,b]) = 1/x(-cos(x)) - integral(1/x^2(cos(x)) dx
  32. K

    I Integrating discs to find the gravitational force of a sphere

    I am attempting to prove Newton's shell theorem. There are multiple solutions to this problem, but I am attempting a solution involving adding up the gravitational force of an infinite number of infinitely small disks that are placed together (the discs facing a point mass "m") to form a...
  33. opus

    Integrating Velocity: Questions on Displacement & Position

    Please see the attached image which are of my notes. In integrating acceleration, I have no confusions. But I have a specific question about integrating velocity. When we integrate velocity, do we get the displacement of ##x##, or do we get it's position at a certain time? I want to say it's...
  34. Cathr

    I Is the Equation for Integrating an Exact Differential Correct?

    Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential. In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ## Or am I getting it wrong?
  35. M

    Integrating Fermionic fields by parts

    Homework Statement Hello there! I have been studying path integral for fermionic fields, and I don't understand one detail. I have to proove that: \int d^4x\int d^4 y\, \overline{\eta (y)}S^+_F(x-y)(i\partial\!\!\!/-m)\Psi (x)=\int d^4x \, \overline{\eta (x)} \Psi (x) Homework Equations We...
  36. 1

    Solving an ODE by the method of Integrating Factors

    1. y' + y = x y2/32. The problem states we need to solve this ODE by using the method of integrating factors. Every example I found on the internet involving this method was of the form: y' + Py = Q Where P and Q are functions of x only. In the problem I was given however, Q is a function of...
  37. Abimbola1987

    A Integrating a Rogowski Coil's Output

    Dear Sirs, I measuring current with a Rogowski Coil and I want to perform numerical integration on the output. Since the output is fluctuating around ground level, the question is: Should I use the absolute values of the output or not? Thank you.
  38. F

    Integrating an exponential function

    Homework Statement Show ##\int_{0}^{1}e_n(x)\overline e_k(x) dx = 1## if ##n=k## and ##0## otherwise. Homework Equations ##e_n(x) = e^{2\pi inx}##. The Attempt at a Solution Consider 2 cases: case 1: ##n=k##. Then ##\int_{0}^{1} e_n(x) \bar e_k(x) dx = \int_{0}^{1} e_n(x)e_{-k}(x) dx =...
  39. Samson Ogaga Ojako

    A Integrating partial derivatives in a field equation

    I am integrating the below: \begin{equation} \psi(r,v)=\int \left( \frac{\frac{\partial M(r,v)}{\partial r}}{r-2M(r,v)}\right)dr \end{equation} I am trying to write ψ in terms of M. Please, any assistance will be appreciated.
  40. ys98

    I Integrating unit vector ρ

    When doing integration such as \int_{0}^{2\pi} \hat{\rho} d\phi which would give us 2\pi \hat{\rho} , must we decompose \hat{ρ} into sin(\phi) \hat{i} + cos(\phi) \hat{j} , then \int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi , which would give us 0 instead? Thanks
  41. hideelo

    A Integrating Gaussians with complex arguments

    The integral I'm looking at is of the form \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right) Where K \in \mathbb{R} and J \in \mathbb{C} The book I am following (Kardar's Statistical Physics of Fields, Chapter 3 Problem 1) asserts that by completing the square this...
  42. J

    Without integrating, show two metal spheres collide in <425s

    Homework Statement Two identical uniform metal spheres of radius 47 cm are in free space with their centers exactly 1 meter apart. Each has a mass of 5000 kg. Without integrating, show that gravity will cause them to collide in less than 425 seconds. [/B] Source: Classical Mechanics, R...
  43. maven sharma

    Integrating operational amplifier

    Homework Statement Calculate the output voltage of the amplifier 2. Homework Equations [/B] https://www.electronics-tutorials.ws/opamp/opamp32.gif The Attempt at a Solution I do not know how to approach a circuit with RC in series to inverting terminal of integrating op amp
  44. M

    MATLAB Integrating Trisurf for 3D Mesh Comparison in Domain D

    Hi PF! I have a 3D mesh generated via the trisurf function, where they each have different node numbers, but are both defined over the same domain ##D##. See attachments for clear image. If the surfaces are ##f1## and ##f2##, I'd like to compute ##\iint_D(f1-f2)^2## where ##D =...
  45. Marcos Domingues

    Integrating an Expression with d(h³)/dt

    Homework Statement [/B] I have this expression: dV/dt = F0 - K*h^(1/2); it describes a variation in time of a fluid volume V in a cone-shaped tank of total volume H*pi*R²/3; By a trigonometric relation we get V = (pi*R²/3*H²)*h³; since tan a = H/R = h/r where: R = radius of the tank; H =...
  46. tarkin

    QM harmonic oscillator - integrating over a gaussian?

    Homework Statement [/B] For the first excited state of a Q.H.O., what is the probability of finding the particle in -0.2 < x < 0.2 Homework Equations Wavefunction for first excited state: Ψ= (√2) y e-y2/2 where: The Attempt at a Solution To find the probability, I tried the integral of...
  47. W

    I Integrating scaled and translated indicator function

    I am struggling to evaluate the following, relatively easy, integral (it might be because its early on a monday morning): $$I_{jk}(a)=\int_0^a\chi_{[0,1)}(2^jx-k)\,dx,$$ where ##\chi_{[0,1)}(x)## denotes the indicator function on ##[0,1)## and ##j,k## are both integers. My idea is to rewrite the...
  48. Jeffsg605

    Integrating the Lift Equation

    I am trying to calculate the lift generate by a helicopter rotor using the lift equation, which is L = \frac{1}{2} \rho V^2 * C_L * S \\ where \\ \rho\mbox{ = density} \\ V\mbox{ = velocity of a point on the rotor} \\ C_L\mbox{ = lift coefficient} \\ S\mbox{ = surface area swept out by the...
  49. J

    How can I integrate sinusoids in python code using FFT?

    Hello, Thank you for taking time to read my post. I have a discrete set of data points that represent an acceleration signal. I want to take the integral of this set of points twice so as to get a function which represents the position over time. To accomplish this, I have taken the FFT of the...
  50. Milsomonk

    Integrating a delta function of a function

    Homework Statement Evaluate the integral: $$\int_{-\infty}^{\infty} dx *\dfrac {\delta (x^2-2ax)} {x+b}$$ Homework Equations $$ x^2-2ax=0 $$ The Attempt at a Solution I know that the delta function can only be none zero when $$ x=2a$$ so then I have the following integral...