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. A

    Integrating Functions with Double Poles: A Contour Integral Approach

    Hi, Can someone tell me how to integrate functions which have a branch point and a pole (of order > 1) on the x-axis. Specifically, I ran into the following problem while playing around with contour integrals, which has a double pole at x = -2 . I tried to do this with a keyhole contour...
  2. A

    Integrating odd functions with infinite discontinuity:

    If an odd function has an infinite discontinuity in its domain, can it be integrated (such that a convergent finite emerges) with that domain included? For example: \int_{-1}^2 \frac{1}{x^{-3}} dx. Intuitively, it can be simplified to \int_1^2 \frac{1}{x^{-3}} dx and thus the infinite...
  3. S

    Integrating Factor and Unintegratable Term

    I was given the following ODE to solve and it seemed simple enough. However, after you have used the integrating factor the integral is not integratable. y' = (1+x^2)y +x^3, y(0)=0 Find y(1) if y(x) is the solution to the above ODE. So I put it in the proper form of: y' + (-1-x^2)y...
  4. H

    Integrating the Cartesian form of Coulomb's law

    Hi, I want to calculate the potential energy between two opposite charges (a dipole) and I know how to integrate Coulomb’s law in the polar form, i.e. in terms of “r” \[...
  5. F

    Integrating with polar co-ordinates

    revising for an exam and have the question let I=∫lim ∞ to 0 e-x2 dx. since x is a dummy variable here, replace it with y to get a second expansion for I. Multiply these two together to get a double integral for I2. Transform into polar co-ordinates noting that dxdy corresponds to r dr d♂. Carry...
  6. T

    LaTeX Integrating Momentum Space: Replacing hslash with d

    does anyone knw the code for how to produce the d slash notation in the integration measure for momentum space? Where (d slash)^n X=(d^n)X/((2pi)^n). Basically all i want to do is replace the h: \hslash with a d.
  7. K

    Integrating e^-4tsint by Parts: A Step-by-Step Guide

    Homework Statement Integrating -e-4tsint Homework Equations Our tutor suggested we did this by integrating by parts in a cyclic fashion and things would cancel out The Attempt at a Solution Taking u = sint t; thus u'=cost And v'=-e-4t; thus (1/4)e-4t \int-e4tsint =...
  8. P

    Integrating a Vector Field Over a Circular Disk

    Hi, How do integrate this? I wish to see it step by step and I'm glad for any help i can get. \int_{ \vec{r}\in{A}} \frac{ \vec{v}+ \vec{\omega}\times\vec{r}}{| \vec{v}+ \vec{\omega}\times\vec{r}|}d^{2}r where A is area of disk with radius R.
  9. F

    Integrating e^(-3x) with u-Substitution

    Homework Statement before we start, i don't know how to do the integral sign, so we'll use [ I need to integrate [ e^(-3x) / 1 + e^(-3x) Homework Equations I've always had trouble with doing integration with e The Attempt at a Solution I used u=1+e^(-3x) du = -3e^(-3x)...
  10. Y

    Integrating factor for solving equation problem.

    Homework Statement Find the gerneral solution of the differential equation below: dy/dt=(-y/t)+2 Homework Equations none The Attempt at a Solution my solution by using integrating factor: 1.find the homogenous solution first dy/y = -1/t dt you get ln(y) = -ln(t) when...
  11. H

    Why Should You Integrate a Power Series from Zero?

    Hello, I have a power series, and a problem with an integration with it, I don't understand why I should integrate it from zero at one point. I have attached a detalied explanation of the problem. http://img79.imageshack.us/img79/5156/powerseriespt4.jpg Any help would be greatly appreciated.
  12. G

    Proof of Fubini's Theorem: Integrating h(x,y)

    What's the proof of this fundamental theorem? Let (X,B,u) and (Y,C,v) be sigma finite measure spaces, Let f in L1(X,B,u) and g in L1(Y,C,v). Let h(x,y)=f(x)g(y). Then, h is in L1(XxY,BxC,uxv) and \int hd(u\times v)=\int fdu \int gdv should be an easy application of fubini,but i really have no...
  13. M

    Integrating x^2 sin x: Find Exact Value of Abs(-pi/2 to pi/2)

    The question asks me to integrate x^2 sin x dx and then use it to find the exact value of the integration of abs ( x^2 sin x dx) with upper limit of pi/2 and lower limit of -pi/2. I have found the integration of x^2 sin x dx which is -x^2 cosx + 2xsinX + 2 cosx + C but after putting in the...
  14. H

    Integrating the minimum metric

    Hello, I tried this in analysis but maybe it is a more topological question. If given a function f on R such that \int_R f(x)dx = 1 and is decreasing and 1-lipschitz, show that the function g(y) = min{x,f(x)} where y = x-f(x) and x>=0, also satisfies \int_Y g(y)dy=1. I really would...
  15. T

    Integrating Half-Loop Radius R: Confused?

    I can't seem to integrate this correctly. I only need the half loop integration, as i have the correct integration for the infinite lines. Homework Statement We start with the top half of a half-loop of radius R, centered at the origin, with infinite line segments traveling in the +/- x...
  16. M

    Integrating x^2/((x^2-1)^(1/2)): A Step-by-Step Guide

    Homework Equations Need to integrate x^2/((x^2-1)^(1/2)). The Attempt at a Solution I first broke the equation into (x^2-1)^(1/2) + (x^2-1)^(-1/2) Hence Integral (x^2/((x^2-1)^(1/2))) = Integral((x^2-1)^(1/2)) + Integral((x^2-1)^(-1/2))) Further on Integral((x^2-1)^(1/2)) is an...
  17. D

    Integrating through singularities

    I am a little bit confused about dealing with integrals around singularities because my professor seems to treat some situations more rigorously than others. We talked this integral and said \int_{-\infty}^{\infty}\frac{1}{x^3} dx = undefined This seems a little bit unintuitive to me...
  18. J

    Integrating sec x dx: Multiply by \frac{tan x + sec x}{tan x + sec x}

    Homework Statement By multiplying the integrand sec x dx by \frac{tan x + sec x}{tan x + sec x} find the integral of sec x dx Homework Equations d/dx sec x = tan x.sec x d/dx tan x = sec^2 x The Attempt at a Solution sec x dx(\frac{tan x + sec x}{tan x + sec x}) => \frac{tan...
  19. E

    Integrating Work from Force(time)

    Hi folks: Question for you guys. I've generalized a question that keeps coming up in class...but that keeps going unexplained. This is a somewhat involved problem which requires integrating force to find work. Thing is, force is always integrated as a function of position NOT time. This type of...
  20. T

    Two questions regarding integrating items

    Homework Statement I have a couple of questions about some intriguing looking equations. Firstly, I have to integrate: \frac{sin\theta}{cos\theta} d\theta Is there an answer to just dividing them? because sin/cos = tan, but that doesn't seem to help? Secondly, I have to integrate...
  21. B

    Integrating Gaussian functions with erf

    Homework Statement I'm doing a problem on Gaussian functions (there are other constants to make it interesting, but I've removed them here): 1. \int_{0}^{x} e^{-x^2} dx 2. \int_{0}^{x} x e^{-x^2} dx 3. \int_{0}^{x} x^2 e^{-x^2} dx We know that erf(z) =...
  22. Z

    How Do You Integrate Over an Octant for \(\frac{1}{x^2+y^2+z^2}\)?

    \int^{1.5}_{0}\int^{1.5}_{0}\int^{1.5}_{0}\frac{1}{x^2+y^2+z^2}dxdydz I tried converting this to spherical and only integrating over a quarter of the octant but with no luck. Can someone please point me in the right direction. Thanks!
  23. J

    Integrating Natural Log Function using Integration by Parts Method

    Integrating Natural Log Function using "Integration by Parts" Method Homework Statement The problem says to integrate ln(2x+1)dx Homework Equations I used u=ln(2x+1); du = 2dx/(2x+1); dv=dx; v=x The Attempt at a Solution So, I integrated it using that (above) 'dictionary' and I...
  24. A

    Integrating seperable equation

    Homework Statement Separable equations dy/dx = y * e^(sinx +cosy) and dy/dx = sin(x^y) The Attempt at a Solution For the first problem, I did dy/dx = y * e^(sinx) * e^(cosy) and separated. However, I can't figure out how to integrate e^(sinx)dx on the right. Did I do somehting wrong? I...
  25. D

    Integrating (e^ax)sin(bx): A Step-by-Step Guide

    How do I integrate (e^ax)sin(bx)
  26. J

    Integrating inverse trig functions

    I was working on this problem \int{\frac{x^3}{x^{2}+1}}dx I at first tried to use one of the inverse trig functions but couldn't get the form to match...should I try to use log properties...making the denominaor u and trying to get the numerator 1?
  27. M

    Integrating the rational fractions

    please... i need a help in integrating the partial fractions i can't proceed to the integration part if i don't understand the patter in finding the constant... that is... if the given is: ʃ ( (x^5+1) / ((x^3)(x+1)) )dx then; ʃ ( x-2 + ( 4x^3+1 ) / ( x^4 + 2x^3) ) ʃ ( x-2 + (...
  28. M

    Integrating irrational functions

    Hello everyone! I was wondering.. if you could help me calculate some integrals: It's not for Homework or something, just my curiosity: \displaystyle{\int}\sqrt[3]{x^2-1} dx What would you suggest? I tried substitution, thou it seems to me useless. Are these integrals common in...
  29. D

    Integrating Exponential Function with Infinite Upper Boundary

    if f(x) = pi*xe^(-x^2) integrating this function if the lower boundary is 0 and the upper boundary is infinity is the answer pi*(2e-1). is this right?
  30. quasar987

    Integrating a 2-form on a 2-cube

    I read, again in Spivak's Calculus on Manifolds, that the integral of 1-form over a 1-cube is equivalent to a line integral. And indeed, if I consider the 1-form w = Pdx + Qdy on R², and c a given 1-cube in R², I find that \int_c\omega = \int_0^1 F(c(t))\cdot c'(t)dt where F=(P,Q), which...
  31. A

    Solving Integrals: A Step-by-Step Guide for Integrating by Parts

    Homework Statement \text {Evaluate } \int^m_1 x^{3}ln{x}\,dx Homework Equations The Attempt at a Solution Integrating by parts, but not sure which term to substitute out...it's not turning out clean...argh I've done every other problem except for this one, can someone just...
  32. D

    Calculating Line Integral on Unit Circle with Complex Analysis Method

    Good old complex analysis. I'm trying to evaluate a line integral which looks like this \ointe (z + [1/z]) for |z| = 1 So I guess I'm dealing with a circle with a radius 1, so I've parameterised: z = eit I need to sub this into my formula of: \intc f(z)dz = \intf(z(t)) z'(t)dt (this is...
  33. R

    How do I solve for the integration of x^2 e^(x^3) without a prefix?

    intergrating "e" I'm doing some intergration q's and I'm stuck on one which involves e [x^2 e^(x^3) ]dx I know to integrate you "add one to the power and divide by the new power.. would that make the solution ((x^3)/3) ((e^(x^4))/(x^4)? hope that makes a bit of sense..
  34. G

    Integrating Newton's equations of motion

    Hello, I have a little project I'm playing with that involves calculating a series of forces and summing them to define the motion of an object (in this case, a walking pedestrian). The force equation is modeled on time-dependent vectors and scalars and is solved using Gear's...
  35. H

    Finding the Integrating Factor

    Hi ok so te question is asking for me to fin the integrating factor of (y2x+y)dy + (x2y+2x)dy = 0 the only thing i know is that the integrating factor should be a function of xy Can some one pleas explain how to do this? Thank you.
  36. R

    Integrating x√(1+x²) Without Trig Functions

    Hi I am trying to integrate x \sqrt{1+x^2}dx by parts...but it seems to involve trigonometric functions - is it possible to solve this integral without using trig functions? Thx
  37. F

    Integrating 1/e^x [ (1-e^-2x)^1/2]

    Homework Statement Integrate. 1/ [e^x (1-e^(-2x))^1/2] Homework Equations integration by parts The Attempt at a Solution First I split up the integral so it would be (1/e^-x)[(1/(1-e^-2x)^1/2] Then I set u=e^-x, dv= (1-e^-2x)^-1/2 du= -e^-x For my v I got 2(1-e^-2x)^1/2 but I...
  38. J

    What is purpose of an integrating factor and how do you get one?

    Homework Statement Am I doing this correctly? Is 2y^4dx +4xy^3dy exact or inexact? Homework Equations The Attempt at a Solution So if its exact, then M(x,y)dx = N(x,y)dy right? (Euler) But how can I take partial derivative of 2y^4 with respect to x, if there is no x to...
  39. H

    Integrating DW/RDR: A Step-by-Step Guide

    Plz explain me how can i integrate dw/rdr
  40. R

    Solving Differential Equations with Integrating Factor

    Hello, I was looking at the differential equation (x^2 + 1) \frac{dy}{dx} + xy = 0. This solution to this equation has to be y = \frac{c}{\sqrt{x^2 +1}} So, when do we usually need to use the method of integrating factor? Can we solve all linear equations (1st order) using this method...
  41. R

    Integrating indefinitely: (ln x)/x^3

    edit: Nevermind, I realized a way to find the answer after posting it. Though I still don't know about the thing involving the notes, can someone confirm if what I have written down as my teacher doing is true? Homework Statement note: I'll use this S as an integral symbol. S(ln...
  42. K

    Integrating the first order rate law

    -d[A]/dt = k[A] - Int( d[A]/[A]) = Int (k dt) - ( ln[A] + ln[A0] ) = kt ln[A] = -kt - ln[A0] Where am I wrong?
  43. L

    How do you set up an integral integrating two functions within a domain?

    How do you set up an integral integrating two functions within a domain?? Homework Statement I have to integrate (2 + x + y) within the domain that is the area between 0 and 1 and (x+y<=1) I know how to integrate well, I think it's a double integral but I'm not really sure what the range...
  44. B

    An ACTUAL post: Integrating exp() over certain range

    An ACTUAL urgent post: Integrating exp() over certain range Hi, Simplified problem: Suppose I have two exponentials \[ e^{ - (x + a - b)} \forall x + a -b> 0 \] \[ e^{ - (x + b)} \forall x + b> 0 \] Then suppose I wanted to integrate: \[ \int\limits_{ - \infty }^\infty {e^{ - (x + b)}...
  45. D

    Integrating Sin(6θ): Am I Close?

    Homework Statement \int\sin(6\theta) d\theta Homework Equations The Attempt at a Solution \int6\cos6\theta Am I close?
  46. P

    Easier way of finding Integrating Factor for Exact Differential Equation?

    Easier way of finding Integrating Factor for Exact Differential Equation?? Homework Statement ,find an integrating factor and then solve the following: [4(x3/y2)+(3/y)]dx + [3(x/y2)+4y]dy = 0 Homework Equations u(y)=y2 is a valid integrating factor that yields a solution...
  47. B

    How can I integrate (1-x)ln(1-x) using integration by parts?

    Hello So I have a problem, which is to use integration by parts to integrate... \int^{1}_{0}(1-x) ln (1-x) dx The way I have been working is it to separate it out into just... \int^{1}_{0}ln (1-x) dx - \int^{1}_{0}x ln (1-x) dx and then integrating by parts on each of these...
  48. B

    Help required integrating this function over all time

    Hello, I am trying to derive a function which describes the probability of two-photons interfering at a beamsplitter however I'm stuck on this particular integral. I need to solve: P(\tau,\delta\tau,\Delta)=\frac{1}{4}\int|\zeta_{1}(t+\tau)\zeta_{2}(t)-\zeta_{1}(t)\zeta_{2}(t+\tau)|^2dt The...
  49. H_man

    Integrating a Monster: Achieving a Solution with Elegance

    \int_{0}^{\pi/2}(cos(\theta)*exp(-2[\pi(1-cos(\theta))]^2)/k^2)/erf(2\pi/k) Where of course the error function erf is defined as: erf(x)=2/\pi\int_{0}^{x}exp(-t^2)dt Anyway... this is the problem I want to integrate. I am not looking for someone to post a solution. My question is...
  50. H

    Numerically integrating the Planck distribution

    I'm working on a project that requires me to numerically integrate the Planck spectral distribution. The object is to find the median wavelength, with exactly 50% of the radiance on either side. I'm using a standard composite Simpson rule method and I get good convergence with temps around...
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