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. T

    Integrating [e^x / (1 + e^2x ) ]dx

    integ [e^x / (1 + e^2x ) ]dx.can someone show me how to solve this by using 1 / (1 + x^2 ) formula?? pls...
  2. W

    How Do You Solve This Tricky Integral?

    I am having a heck of a hard time with this integral... I have tried everything what I can think of: \int \! \left( {e^{x}}+{e^{-x}} \right) ^{-1}{dx} I tried integration by parts... I ended up getting \left( {e^{x}} \right) ^{-1} even thought the right answer, according to Maple and my...
  3. M

    Solving Non-Exact Differential Equations with Integrating Factor

    Hello everyone I understand how to solve exact equations, but what happens when they arnt' exact? I'm confused on what I'm suppose to do! Does anyone feel like explaning hte process to me, if given an integrating factor/> or give me a website? Here is my problem: Check that the equation...
  4. B

    What is the Integrating Factor for x^n*y^m?

    Hey everyone, I need to find an integrating factor of the form x^n*y^m, to solve a differential equation i have... however i do not know the process to solve for an integration of this form.. .any help?? Thanks Steph
  5. S

    Integrating Laguerre Polynomials - Fine structure hydrogen

    Hi I have the following problem: To calculate the fine structure energy corrections for the hydrogen atom, one has to calculate the expectation value for (R,R/r^m), where R is the solution of the radial part of the schroedinger equation (i.e. essentially associated laguerre polynomial) and...
  6. N

    Integrating Problem: Solve Questions Using Formula for \int [f(x)]^nf(x)

    Please help me to solve the following questions using \int [f(x)]^nf(x)=\frac{[f(x)]^{n+1}}{n+1} \int tan {2x} dx
  7. S

    Integrating a Diff. Equation: Seeking Assistance

    Could someone please point me forwards again. By integrating the following equation twice... \frac{1}{x^2}\frac{d}{dy}(x^2 \frac{dx}{dy}) = 0 I tried integrating by parts but came to a sticky end. many thanks skook
  8. T

    Integrating a Complex Integral: Solving the Mystery of the Missing Factor 3

    I'm trying to perform the following integral \pi \int\limits_0^\pi {e^{2x} } \left( {\frac{1}{2} - \frac{1}{2}\cos 2x} \right)dx I split the integral and temporarely ignore the Pi so that I get \frac{1}{2}\int {e^{2x} dx} - \frac{1}{2}\int {e^{2x} \cdot \cos } \left( {2x} \right)dx...
  9. A

    Integrating factor of (a+1)ydx + (b+1)xdy = 0

    for the equation, (a+1)ydx+(b+1)xdy=0, i am wondering how to get (x^a)(y^b) as an integrating factor~ the following is my work: (1/F)(dF/dx)=(a-b)/[(b+1)x] => F=cx^[(a-b)/(b+1)] why doesn't that method work?
  10. A

    Is e^x the Optimal Integrating Factor for Solving Differential Equations?

    for the question, siny+cosydy=0, i want to find an integrating factor. my work: (1/F)(dF/dx)=(1/cosy)(cosy+siny)=1+tany =>lny=x +xtany +c` => y =ce^(x+xtany) however, the question wants the integrating factor to be e^x... why?
  11. R

    Solving ODE: Integrating Factor for Problem 4d

    Howdy, I've read this forum for some time, however this is my first post. I am attempting to solve this ODE. I am looking to find an integrating factor, then solve. I have attached the link to the problem set if my input here is ambiguous. Number 4d. Thank you kindly for any help you might...
  12. M

    Integrating Trig: Solving Definite Integrals with Sin and Cos

    Ok, so we have \int_{0}^{1}\left(\sin{2x}*\cos{2x}\right)dx Using the double angle forumla we change the integrand (1/2)\int_{0}^{1}\left(2*\sin{2x}*\cos{2x}\right)dx which converts to (1/2)\int_{0}^{1}\left(\sin{4x}\right)dx This is where I run into trouble... I'm trying to...
  13. A

    Integrating Factors: Spotting D.E. Need for Factor

    how do you spot that a D.E. needs an integrating factor, besides experience?
  14. I

    Integrating a second derivative-involving solution for Simple Harmonic Motion

    Hello, i am now in the process of integrating m(d^2x/dt^2)=-kx which i know i will have to do twice in order to obtain the general solution to simple harmonic motion, x= Acos(wt+c) c=phi but I'm just having problems with the second derivative of acceleration (d^2*x/dt^2) when it comes to...
  15. V

    Integrating the function 1/x^2. Something I don't understand

    While integrating the function f(x) = \frac{1}{x ^ 2}, I came across something I don't understand: \int \frac{1}{x ^ 2}dx = - \frac{1}{x} + C Let f(x) := \frac{1}{x ^ 2} f(x) > 0, \forall x \in \mathbb{R} \int_{-1}^{1}f(x)dx = -1 - (-(-1)) = -2:confused: Why this happened? :confused: It's...
  16. bayan

    How to Use Integration to Find the Volume of a Rotated Function in Mathematica

    hi felles. I am trying to find what is the volume of the y=\frac{a}{x^2}+b is when it is rotated in y-axis. The values of a is 1 and b is -1. max hight is 3 and min is 0. I was trying to integrade and ended up with V=\frac{-Pi}{y^2+2Y+1} where y is 3. Is this right? I did a U...
  17. G

    Integrating a Tricky Devil: $\int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta$

    \int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta Anyone mind sparing a little hint for this tricky devil? I can't even get started on it. \cot^4(\theta)\cos(\theta) dosn't seem any better either. I've tried using identities but I end up with nastier ones?
  18. RadiationX

    Integrating $\int_0^{\sqrt{6}}e^{-x^2}\frac{x^2}{2}$: U-Substitution or Parts?

    \int_0^\sqrt{6}}e^{-x^2}\frac{x^2}{2} should i use a u-substitution or integration by parts?
  19. B

    Solving "ydx - \left( {x + y^3 } \right)dy = 0" with an Integrating Factor

    Q. By finding a suitable integrating factor, solve the following equation: \left( {x + y^3 } \right)y' = y (treat y as the independent variable). Answer: Exact equation is y^{ - 1} \left( {\frac{{dx}}{{dy}}} \right) - xy^{ - 2} = y leading to x = y\left( {k + \frac{{y^2 }}{2}} \right)...
  20. C

    Integrating exp(-x^2) and some other stuff

    Hello everybody I'm very much interested in the thread about "Feynmans Calculus" (having read the books, too). The problem is I don't understand quite some of the stuff, because I don't have the necessary fundamental knowledge. So I thought to confront you with some lower level questions...
  21. himanshu121

    How do we find Integrating factor for a General Diff equation

    For eg is there a way to find IF for pydx +qxdy +x^my^n(rydx+sxdy)=0
  22. T

    Understanding Integrating Factors and Differential Operators

    I am going to be gone all day tomorrow at a conference track meet and am unable to ask my teacher how to do integrating factors and differential operators. I leave tomorrow at 9:15 am and was hoping to have some examples to take with me to study. If someone could help me walk through a these...
  23. E

    Integrating square to triangle?

    Hi guys, I need a bit of help with this. I've got an op-amp and the standard formula: V_{out} = -\frac{1}{RC}\int V_{in} dt And i need to integrate a square wave from it in order to determine some capacitor/resistor values to get an output amplitude of 5V and freq 200Hz (triangle wave)...
  24. A

    Integrating to find the volume

    Help, I'm trying to find the hypervolume of a hypersphere and I'm stuck on this: V^4= 2(\frac{4\pi} {3}) \int_0^r (\sqrt{r^2-x^2}) ^3 dx I don't know how to do the integration, and I can't expand the (\sqrt{r^2-x^2}) ^3 The answer should be \frac{\pi ^2}{2}r^4 Please help, thanks
  25. C

    Integrating 1/-(x-2) and 1/(x-2) - What To Do?

    Ok, I have to integrate this [int a=0 b=3] 1 / sqrt[abs(x-2)] dx what should i do ? do the improper integral of 1/-(x-2) and 1/(x-2) ?
  26. C

    Integrating in Spherical Co-Ordinates.

    I have the following Integral \int ^1 _0 \int _0 ^\sqrt{1-x^2} \int _0 ^\sqrt{1-x^2-y^2} \frac{1}{1+(x^2)+(y^2)+(z^2)} dzdydx (With the limits working properly!) Converted to spherical Cor-ordinates, I have \int ^\frac{\pi}{2} _0 \int _0 ^\frac{\pi}{2} \int _0 ^1...
  27. K

    How can I find the antiderivative of dx/dv^2 in my physics book?

    Integrating dx/dv^2 ?? i'm trying to figure out an example in my physics book but i don't quite understand the maths. [tex] \int \frac {dv} {v^2} = - \frac {1} {v} [\tex] how does this happen?? looking at the basic antiderivative formulas section in my maths book, it says that...
  28. A

    Can the Volume of a Sphere be Calculated by Integrating the Area of Circles?

    i noticed that if i integrate 2 \pi r i get \int 2 \pi r dr=\pi r^2 i figured its because the area of a circle can be seen as the sum of circumference's of circles with radius 0 to radius r i was thinking if the half volume of a ball also be seen as made from the sum of areas of circules...
  29. D

    Integrating 1/(u⁴+1), 1/(u⁵+1), and 1/(u⁶+1)

    Determine the integral: y = \int_{0}^{1} 1/(u^4+1)du and y = \int_{0}^{1} 1/(u^5+1)du and y = \int_{0}^{1} 1/(u^6+1)du
  30. A

    Integrating Force: Derive Distance L for AP Question

    This last part of an AP questions is giving me some trouble, mostly because i involves integrating and i never took Calculus. Part D: The dart is now shot into a block of wood that is in a fixed place. The block exerts a Force F on the dart that is proportional to the dart's Velocity V and in...
  31. W

    Integrating cos(u^2): A Calc One Challenge

    \int cos(u^2)du Is it doable at a Calc One level? I tried by parts and got to \int cos(u^2)du = ucos(u^2) + 2\int(u^2sin(u^2)du but I am having a brain fart as to hwo to advance, trying again by parts.
  32. RadiationX

    Integrating e^x/(e^{2x} + 1): Long Division?

    i'm having trouble rewriting this integral:\int\frac{e^x}{e^{2x} + 1} so that it will be in the arctan formula: should i use long divison here? if it were not for the e^x in the numerator i'd be fine.
  33. N

    Integrating factor strategy

    Q. Motivate the Integrating factor strategy for U ( "Mew" ) of y I know how to prove it for "Mew" of x but how to do for "mew" of y Maybe something like this. Mdx (x.y) + Ndy ( x, y ) = 0 Assume this is differentiable so let us multiply by "mew" of x on both sides to make it...
  34. M

    Integrating a 3x3 orientation matrix

    Hi there, (I hope this post is in the right forum) I'm trying to integrate a 3x3 orientation matrix using a vector representing rotational velocity (in 3d) This is the formula I'm using: newOrientation = orientation + (dt)(~w)(orientation) where w is the vector rotational velocity...
  35. A

    Help Integrating Tripple Integral for x+y+z=1

    the question is "evaluate \iiint z \,dv, of a solid tetrahedron bounded by the four planes x=0,y=0,z=0, and x+y+z=1" I can set up the problem correctly but i can't seem to integrate it right \int_{0}^1 \int_{0}^{1-x} \int_{0}^{1-x-y} z dzdydx (1/2) \int_{0}^1 \int_{0}^{1-x} (1-x-y)^2dydx...
  36. A

    Integrating Population Differential Equation: Need Help!

    I have this population differential equation dP/dt=k1(P)-k2(P) where k1 and k2 are proportionality constants. I need to integrate and analyze where k1>k2, k1=k2, and k1<k2. Trouble is, I don't think I'm integrating this right. I get P=e^(t+C)(k1-k2). I know this should be easy but I don't think...
  37. D

    Integrating $\frac{1}{\sqrt{2\beta x-\alpha x^2}}$: A Guide

    Does anyone know how to intergrate \frac{1}{\sqrt{2\beta x-\alpha x^2}} I went to wolfram and type it in, but it gave me a weird number.
  38. B

    Integrating a wave function

    Hey. I am pretty confident i have solve this problem. I just solve the integral of the given wave function, with the given limits... However, I am having a difficult time integrating it. The sqrt(2/L) can be brought outside of the integral, but what can i with the sin function? The wave...
  39. J

    How do you determine which denominators to multiply in integrating fractions?

    We just started this and I mostly understand it except when it comes to using A, B, C, etc substitution. What I mean is this, here is an exampe. (6x^2+x+1)/(x^2+1)(x-1) = (Ax+B)/(x^2+1) + (C)/(x-1) You then multiply by denominators so you end up with (Ax+B)*(x-1) + (C)*(x^2+1). You...
  40. C

    Integrating an Improper Divergent Integral & Ellipsoid Volume

    I need help with two questions. Find a divergent improper integral whose value is neither infinity nor -infinity. 2. Find the volume of an ellipsoid (a^2*x^2) + (b^2*8y^2) + (c^2*z^2) = a^2*b^2*c^2 using integration.
  41. M

    Integrating Polar Equations: Online Resources

    Can anyone point me to some online resources on how to integrate polar equations? Thanks.
  42. A

    Can you integrate sin(sqrt(x)) using a trigonometric substitution?

    I need to know how to integrate this function: sin(sqrt(x)) I did this: u = sqrt(x) du/dx = 1/(2sqrt(x)) S(sin(sqrt(x))dx) = S(sin(u)*dx*du/dx/(2sqrt(x)) = S(Sin(u)/u du) But then I got stuck: integration by parts won't work, trig substitution is out... The one thing I did come up...
  43. T

    Integrating and differentiating the number e.

    In calculus, my work has recently involved integrating and differentiating the numer e, of which I am very unsure of how to do. I set up some examples for myself to try to figure out, could anyone tell me if they are correct? Please correct me if I am wrong, or tell me where I have made a...
  44. C

    Why the modulus signs when integrating f'(x)/f(x)?

    I am able to proove to myself, through generalised substitution, that the integral of f'(x)/f(x) is lnf(x)+c, but where do the modulus signs come from? ie - The accepted integral is ln|f(x)|+c, not lnf(x)+c Thanks in advance. :smile:
  45. N

    Need Help - Exact Integrating Factor?

    Here's the problem: y\left( 2x-y-1 \right) dx + x\left( 2y-x-1 \right) dy = 0 So rewrite it as: \left( 2xy-y^2-y \right) dx + \left( 2xy-x^2-x \right) dy = 0 Now it's in the form,…P(x,y) dx + Q(x,y) dy = 0 \frac{\partial P(x,y)}{\partial y} = 2x-2y-1 \frac {\partial...
  46. D

    Integrating tan^2(u)(sec(u))du: Is There a Clean Way?

    whats the integral of tan^2(u)(sec(u))du? i was trying to integrate (x^2)/sqrt(x^2+1)dx, and came into that. it turns out pretty messy though, is there a clean way to do it?
  47. U

    Integrating Force: Understanding \oint and \int

    does the integration of Force equal negative potential energy? \int{F}={-U} Also, what is the difference between \oint and \int?
  48. D

    Integrating csc(x): Easier Than You Think

    haven't found a way of doing it so far. I have a feeling that it's extremely easy, and I'm missing how to do it somehow :/
  49. M

    Integrating a Function 1/x

    Yeah, this questions may be a little elementary for some, but I don't seem to have any sources which would be able to tell me how do i integrate a function 1/x. Any help would be great. : )
  50. N

    Integrating the ALLEE Effect: ds/dt = -a s ln(bs)

    how can i integrate the equation presenting the ALLEE EFFect ds/dt = - a s ln(bs) where a and b are constants
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