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. 1

    Integrating to find distance and time - difficult

    I have been looking for problems to work out for practice and came across one I have no idea how to even start about. Say a box is dropped from a plane, with a parachute attached to it. If we are given two equations for velocity in terms of time, such as Vx(t)=Vo cos(θ) e^(-pt) and...
  2. silvermane

    Is Integration by Parts Applicable for X^3/((e^x)-1) from -infinity to infinity?

    Homework Statement Integrating X^3/((e^x)-1), where we integrate from -infinity to infinity The Attempt at a Solution We thought about it, and we're not entirely sure if integration by parts is applicable here. Is there a table with this function perhaps?
  3. J

    Integrating a Circle: Contour Integration Technique

    Homework Statement integrating a circle ,, my main question is that, can we integrate it by contour integration technique ? and if yes ,, would you please show me how :) or just give me a hint :D Thanks is advance :-) Homework Equations y^2 + x^2 = a^2 where a= r suppose...
  4. D

    Not understanding integrating factorsplease clarify/explain

    Homework Statement Specifically what i don't understand is a situation where you can't get the equation in the standard form of y' + p(x)y = q(x) the integrating factor is e^integ p(x), but what if y is part of a sin or cos, as in this equation: (x+2)sin y dx + xcos y dy = 0 if i...
  5. D

    Can't determine the correct integrating factor (DE)

    Homework Statement Solve (x+2)sin y dx + xcos y dy = 0 by finding integrating factor mu(x) Homework Equations i put equation in standard form to get integrating factor dy/dx + (x+2)sin y / x cos y = 0 but don't i need to get the y out of the sin and cos somehow? so that...
  6. D

    Verify an integrating factor for differential equation stuck

    Homework Statement verify that mu= 1/(y^4) is an integrating factor of: (3x^2 - y^2)dy/dx - 2xy = 0 and use it to solve the equation. Homework Equations (3x^2 - y^2)dy/dx - 2xy = 0 first i want equation in standard form right? dy/dx - 2xy / (3x^2 - y^2) = 0 then mu =...
  7. V

    How does integrating this work?

    question: Consider the gaussian distribution: p(x) = Ae^(-\lambda (x-a)^2) (a) use the equation, 1={\int_{-\infty}^{\infty}} p(x)dx (b) find <x>, <x^2> and \sigma ------------------------------------------------------ a) if i take (x-a) to be u, 1=\int_{-\infty}^\infty...
  8. B

    Integrating around contour (Cauchy)

    Homework Statement The question asks that you prove that \int\frac{sin^{2}x}{x^2}dx = \pi / 2 The integral is from zero to infinity, but I don't know how to add those in latex. Homework Equations Use a contour integral to get around the pole at z = 0. The problem is, I'm really really foggy...
  9. D

    Integrating factor problems (property of exponents)

    Homework Statement my integrating factor for the DE ty' + (t+1)y = t is mu(x) = e^integ (1+(1/t)) = e^(t + ln|t| + c) so does this simplify to this...or not? = e^t + t + c so that DE becomes: ((e^t) + t))y = (e^t) + t) and then after integrating... ((e^t) + t))y = e^t +...
  10. D

    Solving a definite integral without integrating

    Orthonormal basis \big(\frac{1}{\sqrt{2}}, cos(2x)\big) \pi \int_{-\pi}^{\pi} sin^4(x)dx=\frac{3\pi}{4} Solve the integral without integrating. sin^4(x)=\big[\frac{1-cos(2x)}{2}\big]^2=\frac{1}{4} - \frac{2cos(2x)}{4} + \frac{cos^2(2x)}{4} I know I could rewrite cos^2(2x) but then I...
  11. M

    Solving ODE with Integrating Factor Method: (1-x/y)dx + (2xy+x/y+x^2/y^2)dy = 0

    Homework Statement Solve: (1-\frac{x}{y})dx + (2xy + \frac{x}{y} + \frac{x^2}{y^2})dy = 0 The Attempt at a Solution No idea what strategy to use here. Tried using an integrating factor, but no success. A lot of x/y in here makes me think I need to use a substitution, but there's also...
  12. C

    Ordinary DE using integrating factor?

    Homework Statement solve the following initial condition problem x (d/dx) y(x) + 4xy(x) = -8-y(x) y(4)=-6 Homework Equations The Attempt at a Solution first i rearranged xy'+y+4xy=-8 xy'+y(1+4x)=-8 y'+y(1/x+4)=-8/x integrating factor:e^\int(1/x+4) e^(lnx+4x) x+e^4x multiplying...
  13. N

    Find integrating factor and solve the equation 3

    y' + y = e^x ; y(0) = 1 1st, i calculate the integrating factor... u(x) = e^x times the integrating factor with DE... y'e^x + ye^x = e^2x dy/dx e^x + ye^x = e^2x d/dx ye^x = e^2x ye^x = ∫ e^2x dx ...= 1/2 e^2x + C y = 1/2 e^x + C the problem here, i didn't get the...
  14. N

    Find integrating factor and solve the equation 2

    Question: (1 - x - z)dx + dz = 0 let M = (1 - x - z) ; N = 1 dM/dz = -x ; dN/dx = 0 hence, not exact. integrating factor... 1/N[ dM/dz - dN/dx ] = 1[-x] .......= -x e^∫-x dx = e ^ [(-x^2)/2] times integrating factor with DE.. {e ^ [(-x^2)/2] -x e ^ [(-x^2)/2] - z e ^...
  15. N

    Find integrating factor and solve the equation

    y dx + x ln x dy = 0 ; x > 0 my integrating factor is x.. so.. multiply with DE, xy dx + x^2 ln x dy = 0 let M = xy ; N = x^2 ln x dM/dy = x ; dN/dx = x + 2x ln x the problem is.. i didn't get the exact equation after multiply the integrating factor.. I've double...
  16. N

    Differential equation given integrating factor

    Homework Statement Show that given function μ is an integrating factor and solve the differential equation.. y^2 dx + (1 + xy) dy = 0 ; μ(x) = e^xy The Attempt at a Solution let M = y^2 N = (1 + xy) dM/dy = 2y dN/dx = y hence, not exact equation. times μ(x) = e^xy to the...
  17. F

    Integrating csc/cot^2: Need Help with Algebra | Assignment Problem"

    Hey guys working on a problem for an assignment but my algebra is weak regrettably and I need some assistance. Note:: I left the x and dx out for clarity. Homework Statement int csc/cot2 The Attempt at a Solution int csc x/cot2x dx= int csc/cos2/sin2 = int cscsin2/cos2 = int...
  18. M

    Help with integrating the kinematic equations

    Hello, Just wondering if someone can help me make sense of something. I realize it's probably a simple problem, but my math skills aren't the best and I can't see through it. I'm trying to end up with this kinematic equation: X= Xo + Vo(t-to) + 1/2(a)(t-to)^2 And to do this the...
  19. N

    How Do I Integrate Kinematics Problems with Units?

    hello, I'm new to the forums. Can someone help me with integrating kinematics problems? For example velocity= Be^(-rt), where B= 3.00 m/s and r=0.500 s^-1. i don't understand how the integral's unit becomes m (since the integral of velocity is displacement). someone help me! thanks
  20. P

    Integrating the Inverse Exponential-Square Root Function

    Homework Statement \int dx/(e^{x}\sqrt{1-e^{-2x}}) Homework Equations The Attempt at a Solution I have absolutely no idea of how to start the problem, any help is greatly appreciated! thanks!
  21. A

    Integrating wrt a function

    When we integrate a function f(t) with respect to t, we are finding the area under the curve f. Intuitively, this is very clear. What is the intuition behind integrating a function with respect to another function? ex. \int f(t)dg where g is itself a function of t?
  22. B

    Can these two equations be solved for T with limits from 0 to 2pi?

    Can someone help solve these two equations: 1. T[Cos(t)] = ∫[Sin(t-x)Cos(t)]dx the limits are from 0 to 2pi 2. T[Sin(t)]=∫[Sin(t-x)Sin(t)]dx; the limits are from 0 to 2pi Thanks
  23. S

    Integrating exp(k1*cos (t + k2)) from 0 to 2*pi

    Hi I'm trying to compute the following integral (in LaTeX notation; * denotes multiplication) \int_0^{2\pi} exp (k_1 * cos (t + k_2)) d t with k_1 and k_2 being known constants. Furthermore k_2 is between 0 and 2 pi. From Wikipedia [1] I get the following formula \int_0^{2\pi}...
  24. T

    Integrating Along a 2D Line Segment

    I've been trying to evaluate an integral for the last few days now and it really has me stumped. I was hoping that maybe someone here would be able to help me out. So the function, cov(x,x'), is fairly basic. It's called a squared exponential covariance function and it evaluates the covariance...
  25. X

    Integrating V(x)DV(X) in NMOS Transistors: A Mathematical Analysis

    I'm in an electronics course, and the book derives an equation for the current-voltage characteristics of an NMOS transistor. In doing so, it integrate this: \int_{0}^{V_{DS}}V(X)\, dV(X)=\frac{V_{DS}^2}{2}I can see that integrating a function F(X) with respect to F(X) turns out to be the same...
  26. S

    Integrating Ln(√(4-y^2)+2) from -2 to 2

    from -2 to 2, integral Ln(√(4-y^2)+2)dy, how to integral that?
  27. S

    Electric Potential by Integrating Poisson's Equation

    Homework Statement A spherical distribution of charge is characterized by a constant charge density \rho for r<= R. For radii greater than R, the charge density is zero. Find the potential \varphi (r) by integrating Poisson's equation. Homework Equations...
  28. J

    Integrating P along Curve c: A Math Challenge

    Homework Statement So I'm trying to find the integral of P = 2xyz^2 along the curve c which is defined by: x=t^2 y=2t z=t^3 t goes from 0 to 1 So the q says that it is the integral of P dr along c Homework Equations The Attempt at a Solution So I know that this...
  29. P

    First-Order integrating factor of the form f(xy)

    (Moderator's note: thread moved from "Differential Equations") M(x,y) + N(x,y)(dy/dx) = 0 f'(xy) = G(xy)f(xy) where G(xy) = (Nx - My)/(xM - yN) Replace xy with a single variable to obtain a simple 1st order differential equation and find f(xy). I got to: ln|f| = Integral(G(xy)) by...
  30. P

    Integrating exponential functions

    How would you integrate something like xe(x^2) by using a substitution like u=x^2
  31. H

    Integrating Tough Fractions: A Step-by-Step Guide

    http://www.freeimagehosting.net/image.php?9722bd5444.png Link: http://www.freeimagehosting.net/image.php?9722bd5444.png" I've tried to used integration by parts and u substitution and I've also tried just multiplying the fraction by the denominator (6-x)^(1/2) but I am still confused at...
  32. J

    Integrate Factor Diff. Equation | Step-by-Step Solution

    Integrating factor! As promised I'm back with integrating factor differential equation.(x^2 + 1)dy/dx -2xy = 2x(x^2+1) y(0)=1 First put into standard from by dividing thru by (x^2 +1 )dy/dx -2xy/(x^2 + 1) = 2x Integrating factor is given by exp( integral of -2x(x^2 + 1))...
  33. Y

    Calc III Problem Integrating Sq Rt's in Arc Length Formula

    Hey guys, I'm studying for a test in calc 3 tomorrow and have run into a problem. On the practice test we have a problem "Find the length of the curve: r=theta^2, 0≤theta≤pi/2" I know the length of a curve in polar coordinates is int(sqrt(r^2 + (dr/dtheta)^2))dtheta...but when I get to where...
  34. B

    Integrating a charge along a line

    Homework Statement total charge Q = 200[nC] is spread along a line from x=−100[mm] to x=+100[mm] . a) write the dV caused by each dQ=λ dx . b) integrate along the line of charge ; write V as a function of y . c) take the derivative with y , to obtain that component of...
  35. S

    Integrating the Square Root: $\displaystyle \Large \int udv$

    Homework Statement http://img37.imageshack.us/img37/1237/63391287.jpg Homework Equations $\displaystyle \Large \int udv$ = uv - $\displaystyle \Large \int vdu$The Attempt at a Solution can i take this 3 out of the integral as well and make it 3/8 *$\displaystyle \Large \int _0^8 sqrt(64 +...
  36. N

    Integrating Absolute Value in Trigonometric Integrals

    Homework Statement I need to solve the integral \int \frac{dx}{\sin{x} |\sin{x}|}. Homework Equations - The Attempt at a Solution I know it's possible, the solution is -\frac{\cos{x} |\sin{x}|}{\sin^2{x}}, and I probably need to write the absolute as the square root of a...
  37. D

    Integrating for logistic growth model

    Homework Statement Hi, can anyone help me solve a differential equation for the logistic growth model? Homework Equations It reads: M'(t) = M(S-M) + I, where M(t) represents the growth of a biomass. "I" represents immigration (in a coral reef) and there is no breeding...
  38. A

    Integrating $\int\frac{xe^{2x}}{(2x+1)^2}dx$

    Homework Statement \int\frac{xe^{2x}}{(2x+1)^2}dx where "e" is the natural number Homework Equations (none) The Attempt at a Solution I tried many ways to solve this problem, but to no avail. the hint on the book said to use substitution and make u=xe^{2x} and du=2xe^{2x}+e^{2x}dx...
  39. S

    Separation of variables / integrating factor problem

    Homework Statement Find the general solution for this differential equation. dy −2x^2 + y^2 + x dx = x yHomework Equations The Attempt at a Solution dy/dx = (−2x^2 + y^2 + x) / (x y) let y^2 = v dy/dx = v + x dv/dx v + x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / v x^2 => x (dv/dx)...
  40. P

    ODE question - Integrating factor

    Homework Statement Find all solutions of the equation: y' = (2y)/(t.logt) = 1/t, t > 0 Homework Equations Integrating factor I = exp(\intp(x)dx) where y' + p(x)y = q(x) The Attempt at a Solution Hi everyone, here's what I've done so far: Let p(t) = -2/(t.logt) I =...
  41. R

    Integrating Complexity: Solving \int \frac{1}{x^n(1+x^n)^{1/n}} \;\mathrm{d}x}

    Homework Statement An Integral : \int \frac{1}{x^n(1+x^n)^{1/n}} \;\mathrm{d}x} Homework Equations The Standard integrals. The Attempt at a Solution I'm aware that integrals like this become very easy after a clever substitution...but maybe I'm not that clever...
  42. L

    Integrating Vector Fields: Volume vs. Surface

    Homework Statement \int delxA dv = -\oint Axds where A is a vector field Left hand side is integral over volume. Right hand side is integral over closed surface. Homework Equations The Attempt at a Solution Can't understand what Axds means.
  43. I

    Can the floor function be used in integration from 0 to 2?

    Homework Statement Integral [x] - 2[x/2] dx limits are 0 to 2 I am using [] to represent the floor function. Homework Equations The Attempt at a Solution Of course normal integration gives x^2/2 - x^2/2 which gives 0 for all cases, So is it right to assume a floor function...
  44. K

    Integrating $\frac{x}{1 + \sin(x) \cos(x)}$ from 0 to $\pi$

    \int_0^{\pi} \frac{x}{1+\sin(x)\cos(x)} dx
  45. Y

    OMEGA Formula: Integrating to Find the Answer

    I guess people who know can help. I know that the final formula should look like this: (OMEGA) = 4tan^(-1) [(e)/n(1+e^2+n^2)^(0.5)] , where e=W/L, n=2z/L I started with D(omega) = dAcos(theta) p^2 cos (theta) = z/p then i did a bunch of...
  46. B

    Why e-Integral Factor Omits Constant Integral?

    whenever i see that integrating factor for solving a linear differential equation with eint. p(x) dx and then multiplied out in the equation, there seems to be no constant. i tried solving an equation with it the other day, and got an incorrect solution because of it (i think. at least i got a...
  47. J

    Integrating for volume of a hexagonal pyramid

    Homework Statement A) Find the volume of a truncated pyramid with a regular hexagonal base. perimeter of bottom = 1.2 km perimeter of top = 480 m height = 200m B) If density = 1500 kg/m^3 how much work was done in construction? Homework Equations The Attempt at a Solution I...
  48. I

    Integrating over unit circle

    Homework Statement Express f(x,y) = 1/sqrt(x^2 + y^2) . (y/sqrt(x^2 + y^2)) .exp(-2sqrt(x^2 + y^2)) in terms of polar coordinates \rho and \varphi then evaluate the integral over a circle of radius 1, centered at the origin. Homework Equations x = \rhocos\varphi y =...
  49. K

    MATLAB Integrating and Differentiating Polynomials in MATLAB

    Hello, First of all, I am not trying to "spam" subforums. I found out that my thread shouldn't be posted under homework. Anyways, here it is. Integration Let say there's a polynomial, 5x+6 and you want to integrate from 0 to 3 respect to x, how do you input in MATLAB? (I guess you can't...
  50. B

    Integrating a function in infinite space / sphere

    Homework Statement Integrate exp(-3(sqrt(x**2 + z**2 + y**2))) over infinite space [-inf, inf] on xyz Well transforming to spherical coordinates leaves me with the equation at 3.attempts at a..() but here is my problem, how can you equate an integral over an infinite space to a spherical...
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