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. P

    Problem with integrating heaviside function.

    Homework Statement I need to find the integral of the following equation.. integral from 0 to 8 of {(2H[x-0]+2H[x-4])*(x/8)}dx The Attempt at a Solution Im am not sure what the integral of the heaviside function is? is it 1 or 0? Any help would be appreciated! Thanks
  2. P

    Differential Equations : Integrating factors

    If u(x,y) and v(x,y) are two integrating factors of a diff eqn M(x,y)dx + N(x,y)dy, u/v is not a constant, then u(x,y) = cv(x,y) is a solution to the differential eqn, for every constant c.
  3. R

    Need Help Integrating This Tricky Integrand

    Find the length of the curve: r(t) = <2t, t2, (1/3)t3> r'(t) = <2, 2t, t2> From bounds of t: 0 to 1. So length = integral of the modulus of r'(t): Integral of sqrt(t4+4t2+4) I'm just dead stuck on how to attack it. I tried to make it integral of sqrt((t2+2)2), and then just...
  4. P

    Integrating a Complex Math Problem - Need Help!

    Hello I have problem with integrate \int_{0}^{\frac{\pi}{2}}d\phi_1...\int_{0}^{\frac{\pi}{2}}d\phi_{n-2}sin^{2(n-1)}\phi_1...sin^{2(n-k)}\phi_k ...sin^4\phi_{n-2} Please help me.
  5. N

    Why Does the Integration of exp[abs(x)+3]*delta(x-2) from -1 to 1 Equal Zero?

    Homework Statement Integrate exp[abs(x)+3]*delta(x-2) dx, -1, 1 2. The attempt at a solution f(x)=exp[abs(x)+3]*delta(x-2) f(2)=148.4 Integrate exp[abs(x)+3]*delta(x-2) dx, -1, 1 = 0 [b]3. Why is the answer 0?
  6. L

    Integrating double exponential terms in a function

    Hi, this is not really a homework problem, but something that came up during my research, upon trying to integrate some empirical function. This function consists of many terms, but specifically, there is a term containing double exponential functions which is giving me some trouble. If...
  7. C

    Integrating 3 equations together

    There are 3 equations 1/ Continuity equation Av=constant 2/ Bernoulli's equations P+1/2*p*V^2+pgy=constant 3/ Poiseiulle's equation Question1: If a single blood vessle is narrowed by the build-up of plaque so that its inner radius is reduced. How can I use above three...
  8. G

    Integrating factor for first order linear equations uniqueness theorem

    My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...
  9. A

    How can I integrate an acceleration vector in polar coordinates?

    Hi, Say I have an acceleration vector in polar coordinates: a = -30e_r where the unit vector e_r points in the same direction as the Cartesian unit vector j. How can I integrate that vector so that I have the velocity vector in polar coordinates? I know that if I have an acceleration vector...
  10. P

    Integrating (using partial fractions) Apostol Section 6.25 #25

    Homework Statement \int\frac{4x^5-1}{(x^5+x+1)^2}dx Homework Equations This is in the section on Partial Fractions. The main idea in this section was that you get the integral down to a sum integrals of the following forms: \int\frac{dx}{(x+a)^n} , \int \frac{x dx}{(x^2 + bx + c)^m} ...
  11. D

    Help! Integrating Volume Flux Across a Tube

    Please Help! Integration! Hi! I'm having a little problem with an exercise and I don't know how to sort it out. Basically what I have is a tube of 5mm ejecting spray fuels. At the exit plane I have positioned lasers to measure the volume flux at different locations, starting from one side...
  12. F

    Integrate y''/y: Solving with Logs

    Homework Statement How would you find the integral of y''/y (with respect to x)? Homework Equations The Attempt at a Solution I have absolutely no idea how to begin. All I know is that if it were y'/y, it would be log(y) + c. Perhaps the integral here has something to do with...
  13. D

    Is there a clever method for integrating over asymetric paraboloids

    There are a few questions on the forum about calculating the volume enclosed between an upwards-opening and a downwards-opening paraboloids, and I think I understand the method there. However they all involve symmetric paraboloids, and the intersection of the pair is always contained within a...
  14. A

    Integrating the Gegenbauer function (1st kind) wrt theta

    Homework Statement Find ∫ In(ζ) dθ , where ζ= cosθ and In(ζ) is the gegenbauer function of the first kind. The original problem is to find find ∫ sinθ *I//n(ζ) dθ where I//n= 2nd differential of In(ζ) with respect to ζ. Homework Equations d/dζ In(ζ)=-Pn-1(ζ), where Pn(ζ) is the legendre...
  15. J

    Need help integrating a function involving e^(-t^2)

    Homework Statement The context of this question is physics related but the problem I am having is purely mathematical. g(t)= e^(-(a^2)(t^2))*e^(iwt) (a and w are just constants, i is sqrt(-1), not a constant) I need to integrate this function with respect to t from -infinity to...
  16. L

    Mathematica Plotting and Integrating difficult functions/Maple or mathematica

    Hi Everybody, So basically I spent the summer working on some optics related stuff and now they want me to present my research work. This means I'm going to have to do numerical simulations for diffraction of Laguerre-Gaussian beam modes. In a word, the integrals are ridiculously long and...
  17. P

    Integrating Using Trigonometric Substitution

    Homework Statement Use the method of trigonometric substitution to evaluate the following: \int\frac{x^{2}}{\sqrt{4-9x^{2}}} Homework Equations The only relevant equation that I could think of for this one was the trig identity: sin^{2}\vartheta + cos^{2}\vartheta = 1 The...
  18. A

    Integrating Complex Exponentials

    The integral of a complex exponential ( e^(ix) ) over x from 0 to infinity is supposedly such that the value of the definite integral at the upper limit is zero and so it's just -1/i. Why is this? It's just an oscillating function after all.
  19. Z

    Integrating x * (cos x)^ n between pi/2 and 0

    Would anyone be able to help me do this? I have tried by parts, but did not make progress. As n gets large, the area gets smaller. Your help is appreciated. Z
  20. Telemachus

    Integrating $\nabla \times{F}$ - Finding the Error

    Homework Statement Hi there. I was trying to solve this problem, from the book. The problem statement says: Integrate \nabla \times{F},F=(3y,-xz,-yz^2) over the portion of the surface 2z=x^2+y^2 under the plane z=2, directly and using Stokes theorem. So I started solving the integral...
  21. N

    Integrating a dot product inside an exponential

    Homework Statement This is from Peskin & Schroeder p. 14 in case anybody's interested. The function is U(t)=\frac{1}{(2\pi)^3}\int d^3p\, e^{-it \sqrt{p^2+m^2}}e^{i\vec p\cdot(\vec x-\vec x_0)} Homework Equations The Attempt at a Solution Essentially you write out the dot product as p\cdot...
  22. R

    Integrating infinitesimal conformal transformations

    While it's pretty easy to derive the infinitesimal version of the special conformal transformation of the coordinates: x'^{\mu}=x^{\mu}+c_{\nu}(x^{\mu} x^{\nu}-g^{\mu \nu} x^2) with c infinitesimal, how does one integrate it to obtain the finite version transformation...
  23. B

    MATLAB Integrating a Function of Two Variables in MATLAB

    Hey guys, I have this function: f(r,z) = r*(1 + g(r,z)) The function g(r,z) is a modification of the student t-distribution in z, where the degrees of freedom depend continuously on r. I would like to integrate this function f(r,z) from 0 to R with respect to r. Unfortunately...
  24. J

    Problems Integrating an Infinite Series From E&M

    Hi, guys. For an E&M/quantum mechanics problem I have to integrate the series below: Homework Statement Integrate \int{r^{2}e^{y r} \, dr}.2. The attempt at a solution Using integration by parts and and "differentiating under the integral" give the same answer: I= \frac{2...
  25. B

    Setting up double integral for polar coordinates and integrating

    Link: http://imageshack.us/photo/my-images/39/18463212.jpg/ This is a very long problem so I drew it to make things simpler. Part a) tells me to set up a double integral in polar coordinates giving the total population of the city. I have the following: 2π...4 ∫...∫ δ(r, θ) r dr...
  26. C

    What Is the Best Method to Integrate dx/(x*sqrt(9+16x^2))?

    Homework Statement indefinite integral: dx/(x*sqrt(9+16x^2)) Homework Equations Trig. Substitutions or parts?? The Attempt at a Solution I tried using integration by parts but its got pretty messy...it also resembles a tan trig substitution, but it's within a square root. I'm...
  27. N

    Integrating f(x,y,z): Confirm Ranges for x, y, z

    Homework Statement f(x, y, z) = x^2 ; G is tetrahedron bounded by the coordinate planes and the plane octant with equation x + y + z = 1 ∫ ∫ ∫ x^2 dzdydx I try to set up the ranges for x, y and z.. x+y+z = 1 z = 1-x-y...set the limits for z from z=0 to z = 1-x-y x+y+z = 1 if, z = 0, y =...
  28. S

    Solving an equation after integrating

    Homework Statement i have an equation that i want to solve for b: \int _0^1 (a+b x)^2 x^c dx=2 Homework Equations \int _0^1 (a+b x)^2 x^c dx=2 given: c>0 The Attempt at a Solution To evaluate the integral on the left hand side, I expanded the bracket as: (a+b...
  29. N

    Finding the integrating factor

    I am having trouble with the below: [ 4* (x^3/y^2) + (3/y)] dx + [3*(x/y^2) +4y]dy=0 I found My= -8x^3y^-3 - 3y^-2 and Nx= 3y^-2 i then subtracted Nx from My and divided by [3*(x/y^2) +4y] [-8x^3y^-3 - 6y^-2] / [3*(x/y^2) +4y]. can you guys give me a hint as to where my error is?
  30. A

    Integrating Factor and absolute value

    Hi, I have a general question regarding the integrating factor of first-oder linear DEs. All textbooks that I've seen (which aren't too many) simply drop the absolute symbol when the factor has the form exp(ln(abs(x))). This would evaluate to abs(x), yet the books use simply x. Why is that...
  31. B

    Integrating {C_\ell }: An Overview

    Homework Statement consider the integral, {C_\ell } = \frac{{2\ell + 1}}{{2{j_\ell }(kr)}}\frac{1}{{{2^\ell }\ell !}}\int_{ - 1}^{ + 1} {\frac{{{d^\ell }({{({x^2} - 1)}^\ell })}}{{d{x^\ell }}}{e^{{\bf{i}}krx}}dx} how do you do it? Homework Equations l = integer, as in the...
  32. Telemachus

    Integrating Fourier Series for a Cubic Function

    Homework Statement Hi there. I wanted to intagrate the Fourier series for g(t)=t^2 to get the Fourier series for f(x)=t^3 So I thought making something like: f(t)=3\int_0^t x^2 dx I know that g(t)=t^2\sim \frac{p^2}{3}+\sum_{n=1}^{\infty}\frac{4p^2(-1)^n}{n^2\pi^2}\cos\left (\frac{n\pi...
  33. F

    Integrating Factor, how do you get this?

    Homework Statement My book shows that http://img845.imageshack.us/img845/4875/unleduot.jpg and then they arrived at the results [PLAIN][PLAIN]http://img202.imageshack.us/img202/5442/unleddq.jpg So my problem is that, how exactly did they drop the mu in the first picture?
  34. R

    Integrating Partial Derivatives

    Homework Statement Find the general function f(x,y) that satisifes the following first-order partial differential equations \frac{df}{dx}=4x^3 - 4xy^2 + cos(x) \frac{df}{dy}=-4yx^2 + 4y^3 The Attempt at a Solution I integrated both to get: x^4 - 2x^2y^2 + sin(x) + c(y) and -2y^2x^2 + y^4...
  35. F

    Constant and a variable in a squareroot needs integrating

    Homework Statement http://s39.photobucket.com/albums/e178/dorlomin/enviroment/?action=view&current=core2.jpg http://s39.photobucket.com/albums/e178/dorlomin/enviroment/?action=view&current=core2.jpg" Homework Equations In the question shown the equation is required to be integrated to...
  36. K

    Integrating surface areas with dS and dA

    Homework Statement ζζs yz ds S is the part of the plane x+y+z=1 that lies in the first OctantHomework Equations ζζs f(x,y,z) dS = ζζd f(x,y,g(x,y)) sqrt(1+(df/dx)²+(df/dy)²) dAThe Attempt at a Solution turned dS into dA, and attempted many different regions. Always got an area that was 4-6...
  37. Femme_physics

    Integrating to solve kinematics problem, bad or good?

    My lecturer told me that it's best not to integrate when you're a new student to kinematics because it's important to understand the principles of operation-- which is why we're not using it. On the other hand, I keep being told by this forum that calculus is the true way to fathom mechanics...
  38. B

    When integrating by trig substitution why do you use what you use?

    when integrating by trig substitution why do you use what you use?? for example int. (1+x^2)^0.5 dx why do you use x= tan u i mean obviously because it works, but if you didn't know it works how would you figure it out? i would think that you should use x=sinh u but I've been trying...
  39. A

    How Can I Understand the Integration Process of this Fraction?

    I came across this problem + solution: http://imageshack.us/m/715/2962/inteyk.png but I don't understand the calculus there. How can you integrate the stuff inside the brackets but not integrate that fraction?
  40. L

    Integrating e^x^2: Proof & Explanation

    I went through several threads about the integral of e^x^2 but I could not find anything about the proof why e^x^2 cannot be integrated as an elementry function. Could you explain why?
  41. A

    Help Needed: Integrating a Complex Expression

    I need some help. I don't know how to make the intgral symbol, but here's the question. If somebody can get it started it'd be awesome. 1to e^10 1to e^6 1to e^2 1/xyz dxdydz
  42. W

    Integrating a Function with a Variable and Function Multiplied Together

    Homework Statement http://img857.imageshack.us/i/no34.jpg/ Homework Equations The Fundamental Theorem of Calculus, i.e., taking the derivative of an integral yields the original function. The Attempt at a Solution I am not sure how to go about integrating this function because I...
  43. W

    Numerically Integrating a Damped-Oscillating Mass System

    Hi all. My first time posting. Hopefully it will go well. :) For my ME Lab 1 class I need to numerically differentiate LVDT data to find acceleration of an damped-oscillating mass system and I need to numerically integrate accelerometer data to find displacement of the same...
  44. J

    Confused- Integrating a vector field along a curve in 3D.

    Homework Statement Let f be a vector function, f = (xz, 0, 0), and C a contour formed by the boundary of the surface S S : x^2 + y^2 + z^2 = R^2 , x ≥ 0, y ≥ 0, z ≥ 0 , and oriented counterclockwise (as seen from the origin). Evaluate the integral (Closed integral sign) f · dr , directly as...
  45. C

    Exact Differential Equations and Integrating Factor

    Homework Statement "Show that each of the given differential equations of the form M(x,y)dx + N(x,y)dy = 0 are exact, and then find their general solution using integrating factors μ(x) = e∫h(x)dx and μ(x) = e∫g(y)dy Homework Equations (x2 + y2 + x)dx + (xy)dy = 0The Attempt at a Solution...
  46. F

    Integrating (2rx - x2)1/2 w.r.t. x and Constant r

    Homework Statement Integrate (2rx-x2)1/2 with respect to x where r is a constant. Homework Equations The Attempt at a Solution Thanks
  47. K

    Integrating Electric Displacement

    Suppose the dielectric material is fixed in position and filling the capacitor, and you would have this term in the way of calculating something. \int\nabla\cdot\left[\left(\Delta{D}\right){V}\right]{d}\tau where D is the dielectric displacement Now that turns into (by divergence...
  48. Z

    Integrating a Definite Integral with a Dummy Variable

    If we have a definite integral given by f(x) = ∫sintdt with limits from 0 to x, can we integrate the function directly by substituting t as x because the variable involved in the integrand is a dummy variable?
  49. E

    How do I solve this Bessel function integral using a u-substitution?

    Homework Statement This is part of a vibrating circular membrane problem, so if I need to post more details please let me know. Everything is pretty straight forward with the information I'll provide but you never know. We haven't really learned what these are, just that they are complicated...
  50. L

    Harmonic Functions and Integrating Along Curves: The Role of 90 Degree Rotations

    If f is a smooth function defined on the plane then the integral of its gradient over any closed curve is zero. What about the integral of the 90 degree rotation of the gradient? In symbols, if grad f = (df/dx , df/dy) its 90 degree rotation is (-df/dy , df/dx)
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