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.

View More On Wikipedia.org
Recent contents Top users
  1. C

    Integrating a x^k ln(x) Function with Gamma Function

    Homework Statement "Show that - \int^1_0 x^k\ln{x}\,dx = \frac{1}{(k+1)^2} ; k > -1. Hint: rewrite as a gamma function. Homework Equations Well, I know that \Gamma \left( x \right) = \int\limits_0^\infty {t^{x - 1} e^{ - t} dt}. The Attempt at a Solution I've tried various substitutions...
  2. B

    Integrating Factor Homework: y'+ (2/t)y = (cost)/(t^2), y(pi)=0

    Homework Statement y' + (2/t)y = (cost)/(t^2), and the following condition is given: y(pi) = 0Homework Equations The Attempt at a Solution After employing the integrating factor, I find the solution to be: y=e^{-2t} \int e^{2t} \frac{\cos(t)}{t^2} dt. Evidently, this simplifies all the way to...
  3. J

    How do I rearrange and integrate dx/dt=k(a-x)(b-x) in order to graph x and t?

    Homework Statement I need to integrate dx/dt=k(a-x)(b-x) and then rearrange to find xThe Attempt at a Solution 1/((a-x)(b-x))dx = k dt Integrating dx using partial fractions: 1/(b-a)×(ln((a-x)/(b-x))=kt+c when t=0 x=0 ∴c=(ln(a/b))/(b-a) then when I rearrange I get...
  4. M

    Understanding the Behavior of Trigonometric Functions on Specific Intervals

    I have a question regarding what I highlighted in the paint doc. How can they assume that the function is increasing? As a matter of fact its not increasing on the entire interval from 0 to pi.
  5. Mr-R

    Understanding the Work Needed to Escape a Gravitational Pull

    Hello everyone, What does it mean if I integrate Newton's law of universal gravitation with respect to r. F= GMm/r^2 become 3GMm/r^3 . Is this the work needed to escape a gravitational pull ? Thank you
  6. J

    Mathematica Mathematica: Trouble accurately integrating highly-nonlinear DE

    Hi, I'd like to numerically solve the IVP: x^2 y y''+2x y y'+2 y^2+xy'-x^2 y^3-(x y')^2-y=0,\quad y(x_0)=1,y'(x_0)=0 around the unit circle, x=e^{it}. When I attempt to solve it around the entire circle, I think the integration is veering of course. I believe the solution is single-valued...
  7. K

    Integrating a product of two functions - one lags the other

    Integrating a product of two functions - one "lags" the other I am wondering if there is a way to integrate the following function without first expanding the brackets: \int\limits_{x1}^{x2} x^2\left(x-a\right)^2\,dx The idea behind the question is a bit more complex than I am letting on...
  8. C

    Integrating a normal density to find a CDF

    Homework Statement Let X ~ norm(5,10). Find P(X>10).Homework Equations f(x) = \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} F(x) = P(X<x) = \int_{-∞}^x f(u) du The Attempt at a Solution P(X>10) = 1 - P(X<10) P(X<10) = \int_{-∞}^{10} \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} dx =...
  9. W

    Integrating the time-indepdent Schrodinger equation

    Homework Statement A particle of mass m is confined to move in a one-dimensional and Diract delta-function attractive potential V(x)=-\frac{\hbar^2}{m}\alpha\delta(x) where \alpha is positive. Integrate eh time-independent Schrodinger equation between -\epsilon and \epsilon. Let...
  10. C

    Integrating xe[itex]^{x}[/itex] without integration by parts

    Homework Statement Use the method of separation of variables or an integrating factor to find a particular solution of the differential equation that satisfies the given initial condition. y'=x-y+2 ; y(0)=4 2. The attempt at a solution I've used an integrating factor of e^{x} to...
  11. C

    Integrating Fractions with Substitution

    So the problem is ∫(6x+5)/(2x+1)dx. I know the proper way to solve this is to long divide these two expressions and then solve. However, I tried doing it with substitution. u = 2x+1 dx = du/2 I then reasoned that 3u + 2 = 6x+5 since 3(2x+1) + 2 = 6x+3+2 = 6x+5 so I substituted it on top...
  12. MarkFL

    MHB Speeeed's question at Yahoo Answers regarding special integrating factors

    Here is the question: Here is a link to the question: Show that the given equation is not exact and find the right integrating factor to make this equation exact? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  13. Mandelbroth

    Integrating derivatives of various orders

    I'm trying to figure out the general solution to the integral ##\int \frac{d^ny}{dx^n} \, dy##, where n is a positive integer (Meaning no fractional calculus. Keeping things simple.). So far, I have been working with individual cases to see if I can establish a general pattern and then try a...
  14. R

    Integrating the product of a step function and a trigonometric function

    Homework Statement Integrate -π∫π H(t-π/2)*sin(2t)dt Homework Equations See above. The Attempt at a Solution I can rationalize the slightly simpler integral for the same limits of H(t)*sin(2t) as coming out to 0 due to the definition of the unit step function, but I'm wondering...
  15. Fernando Revilla

    MHB Integrating factor (Madmen's question at Yahoo Answers)

    Here is the question: Here is a link to the question: Differential equations math question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  16. E

    Integrating a natural log in double integral

    Homework Statement \int^{1}_{0}\int^{e^x}_{e^-x}\frac{lny}{y}dydx The attempt at a solution So I am integrating ln(y)/y and I tried it by parts, first with u = ln(y), dv = 1/y, and therefore du = 1/y, and v = ln y but if I use that I get (ln(y))2-\int\frac{lny}{y} again. So I tried...
  17. Z

    Integrating directly from a pair of diff.eqs. without solving them

    This is actually related to a post I made earlier in the differential equations forums, but I've since realized that solving the equations themselves is not necessarily the best way to get where I want to go. Perhaps it's better suited to this forum, since it is an integration problem that I...
  18. Z

    Is there a way to determine if an integral exists for a non-elementary function?

    I've been trying to figure out how to integrate 1/(x+ln(x)) but am not getting anywhere. Mathematica can't do it, and I haven't found it in lists of integrals. Does anyone know if this integral exists in closed form? Same goes for (x+ln(x))/(1+x+ln(x)) Thanks! zeroseven
  19. F

    Integrating Gaussian integral by parts

    Homework Statement We define I_{n} = \int_{-∞}^{∞}x^{2n}e^{-bx^{2}}dx, where n is a positive integer. Use integration by parts to derive:I_{n}=\frac{2n-1}{2b}I_{n-1} Homework Equations Parts formula. The Attempt at a Solution So I'm just stuck here, I'm baffled and confused. Firstly if I...
  20. C

    Integrating a function from 0 to infinity correctly?

    Homework Statement I am trying to integrate the PLanck function to get the Stefan Boltzmann law. After factoring out constants, and substituting x = hv/kT I am left with the following integral: B(T) = ∫ x3/(ex - 1) dx integrated from 0 to ∞ The next step in my notes is that the result...
  21. M

    Integrating to find the volume of sphere

    Homework Statement Hi I have a question about integrating to find the volume of a sphere but before that i need to tell all other results i got or it will not make sence. 1. I needed to show how you get the formula for finding the volume of a sphere by the help of a cirkle with radius r . I...
  22. G

    Electrical Circuit Question - Integrating Amplifier

    Homework Statement http://img801.imageshack.us/img801/4227/capturebsw.png Homework Equations The Attempt at a Solution Alright I'm not really sure how to solve this problem. I have redrawn the circuit below without the operational amplifier. In ideal operation amplifiers there's no...
  23. I

    Integrating exp(-i(px+qy)) dxdy over all space

    In my diffraction notes, this integral comes up on the page about Babinet's principle: \int ^{y=\infty}_{y=-\infty} \int ^{x=\infty}_{x=-\infty} exp(-i(px+qy)) dx dy = \delta (p,q) I'm not sure how this integral is derived as carrying out the integration and putting in the limits seems to...
  24. H

    Integrating normal from 0 to Inf and finding the third moment

    This is (perhaps) a tricky question regarding the moment of normal distribution. Let f(x) be the pdf of normal distribution with mean (-σ^2/b) and variance σ^2, where b is just a constant. The goal is to solve the integral ∫ x^3 f(x) dx integrating from 0 to ∞. I am stuck. Any...
  25. C

    Integrating gravitational acceleration of particles for simulation

    I've tried out the two body problem and tried to work out the trajectories with respect to their center of mass frame(located at the origin) as follows (it worked!:smile:): particles = m1 and m2 Fm1 = force of m2 on m1 Fm2 = force of m1 on m2 a = Gm2\hat{r} / r2 \hat{r} is the unit vector...
  26. S

    Integrating cos^6(x) using cos^4θ - sin^4θ and cos^4θ + sin^4θ

    Using cos^4\theta - sin^4\theta = cos2\theta and cos^4\theta + sin^4\theta = 1 - \frac{1}{2}sin^22\theta evaluate: (i) \displaystyle \int_0^{\frac{\pi}{2}} cos^4\theta \ d\theta adding the two identities given I get: 2cos^4\theta = cos2\theta + 1 - \frac{1}{4} + \frac{1}{4} cos4\theta...
  27. C

    Integrating the sqrt of a function

    Homework Statement Apologies for the vague title, I'm not reall sure what I'm look at here! I am doing some revision on solving the Friedmann equations, and in a lot of cases I end up having to integrate a function that looks like (xan - 1)-1/2 da = dt where a is a function of t, x is a...
  28. S

    Integrating a Fraction using the product rule

    Homework Statement Find the integral of This was the question. There is a way to do it by long division but I am confused with Long division. Instead I tried to do by the method below but I failed... Homework Equations NoneThe Attempt at a Solution I thought maybe I could reduce the...
  29. A

    Integrate by Parts: arctan(√x)dx Explained

    Homework Statement ∫arctan(√x)dx. Using the substitution √x=t: ∫arctan(√x)dx = ∫arctan(t)dt2 This is what I've got written in a solution manual. I don't see why the dt would be squared. Could anyone care explaining me? thanks
  30. D

    Integrating factor for a 2nd order homogeneous linear ODE

    Homework Statement Consider the general linear homogeneous second order equation: P(x)y'' + Q(x)y' + R(x)y = 0 (1) We seek an integrating factor μ(x) such that, upon multiplying Eq. (1) by μ(x), we can write the resulting equation in the form [μ(x)P(x)y']' + μ(x)R(x)y = 0...
  31. S

    Integrating the product of a real and a complex exponential

    Homework Statement \Psi(x,t) = \int^{\infty}_{-\infty} C(p)\Psi_{p}(x,t) dp is a solution to the Schroedinger equation for a free particle, where \Psi_{p}(x,t) = Ae^{i(px-Ept)/\hbar}. For the case C(p) = e^{-(p-p_{0})^{2}/\sigma} where \sigma is a real constant, compute the wavefunction...
  32. J

    Integrating Using Partial Fractions

    Homework Statement This is an arc length problem in three dimensions. I was given the vector r(t)=<et, 1, t> from t=0 to t=1 Homework Equations Arc Length= \int |\sqrt{r'(t)}| dt from t1 to t2 where |\sqrt{r'(t)}| is the magnitude of the derivative of the vector The Attempt at a...
  33. W

    Integrating Without a Calculator: Finding Length of Curve

    Homework Statement Find the length of the curve between x=0 and x=1. Note: can this be done without a calculator?Homework Equations y = sqrt(4-x^2) The Attempt at a Solution x=2sin∅ dx = 2cos∅ d∅ sqrt(4-4(sin∅)^2) ---> 2cos∅integral (0 to 1) sqrt(1+(-2sin∅)^2) integral (0 to 1)...
  34. C

    Why is the output of the integrating op amp reversed?

    Hello, I'm having problem understanding the output of the integrating op amp. https://www.dropbox.com/s/djpg0uhi465ot86/download2.png https://www.dropbox.com/s/djpg0uhi465ot86/download2.png https://www.dropbox.com/s/p9bp9tlyykhjtvf/download.png...
  35. L

    Am I integrating this right: (x^2 + 3x + 11)/(x+2)^4 ?

    Am I integrating this right: (x^2 + 3x + 11)/(x+2)^4 ? 1. ∫(x2 + 3x + 4)/(x+2)4dx 2. With these sorts of problems, I think about integration by partial fractions. So in this case, the denominator factors are all the same, so I have to make each one with different exponents. I wrote...
  36. M

    Swapping Integrals and Sums: When is it Justifiable?

    when using the reimann integral over infinite sums, when is it justifiable to interchange the integral and the sum? \int\displaystyle\sum_{i=1}^{\infty} f_i(x)dx=\displaystyle\sum_{i=1}^{\infty} \int f_i(x)dx thanks ahead for the help!
  37. C

    Can You Simplify Integrals by Separating Radicals?

    Homework Statement for the following integrals, am I allowed to break them up like so: 1. ∫(1)/(sqrt(16-9x²)³) dx = ∫(1)/(√16)³ · ∫(1)/(√-9x²)³ dx 2. ∫(x²)/(sqrt(x²-9)) dx = ∫(x²)/(√x²) · ∫(x²)/(√-9) dx 3. ∫(1)/(x²(sqrt(a²+x²))) dx = ∫(1)/(x²) · ∫(1)/(√a²) · ∫(1)/(√x²) dx...
  38. PhizKid

    Integrating Volume via Shells: What Went Wrong with the Bounds?

    Homework Statement Volume of the region bounded by 1/x^4, y = 0, x = 2, and x = 6 about the axis y = -4 Homework Equations The Attempt at a Solution The height for any shell is x(y) - 2, or \sqrt[4]{\frac{1}{y}} - 2 The radius of any shell is y + 4 So the circumference...
  39. E

    Find the integrating factor u(x) for this exact differential equation

    Homework Statement (3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0
  40. M

    Integrating over piecewise functions

    given a function f(x) that is piecewise smooth on interval -L<x<L except at N-1 points, is \int_{-L}^L f'(x)dx legal or would i have to \sum_{i=1}^N \int_{x_i}^{x_{i+1}} f'(x)dx‎‎ where x_{N+1}=L and x_{1}=-L also, am i correct that if f(x) is piecewise smooth, then f'(x) is piecewise...
  41. P

    Differential equation parametrisation with integrating factor

    Homework Statement Use parametrisation first, derive the equation including y and p = \frac{dy}{dx} and use the integrating factor method to reduce it to an exact equation. Leave the solution in implicit parametric form. (y')^{3} + y^{2} = xyy'The Attempt at a Solution I'm really lost at...
  42. MarkFL

    MHB Fishfood12m's question at Yahoo Answers regarding special integrating factors

    Here is the question: Here is a link to the question: Integrating Factor Help Please!? - Yahoo! Answers I have posted a link there so the OP can find my response.
  43. W

    How Do You Integrate the Rational Function ∫((x^2+x-7)/(x+3))dx?

    Homework Statement Integrate the following: Homework Equations ∫(x^2+x-7)/(x+3) The Attempt at a Solution The only way I can think of solving this would be to split up each term into a separate fraction.
  44. W

    Integrating Rational Functions: Solving ∫(x/(x-1)^3)

    Homework Statement integrate the following:Homework Equations ∫(x/(x-1)^3The Attempt at a Solution i've tried u-substitution, finding an inverse trig function that matched the formula, and still can't figure out how to solve this problem. u-subtitution for u=x gives the same problem...
  45. S

    Integrating a differential form on a manifold without parametric equations

    I am sure that this can be done, but I haven't been able to figure it out, Is there a way to integrate a differential form on a manifold without using the parametric equations of the manifold? So that you can just use the manifold's charts instead of parametric equations? If you a function...
  46. B

    Integrating the Dirac Delta function

    Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...
  47. P

    Integrating Acceleration for Distance

    I'm trying to integrate acceleration from an acceleraometer to find a distance travelled. I have heard all the stories about this not being accurate but i didn't come up with the method I'm just trying to implement an algorithm to do it. I'ts justan estimate for wear rates, not positioning...
  48. D

    Integrating a concentration profile

    Hello- I have a diffusion profile, in which I plot the decrease in concentration versus distance of my sample. I am trying to find the increase in mass of the overall sample. How would I do this? I have integrated the function and am wondering what type of information that would supply...
  49. DiracPool

    Integrating 1/x: From Undefined to Natural Logs

    The integral of 1/x is ln(x). Where does that come from? That always puzzled me. We can continue to take derivatives through x^0 and into the negative integers, and just use the plain old power rule to get the answers. We can do the same for the integral of x all the way from negative...
  50. R

    Integrating complex exponentials with Infinite limits

    Hi all, I'm trying to integrate the function below with respect to x exp(ix)-exp(-ix) With infinity and negative infinity as the limits. Would the integration be possible?
Back
Top