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

    Differentiating and Integrating the Lambert W function

    Here was my thinking for differentiation (which, by the way, is wrong): By the definition of the function, the following equations are equal: $$W(xe^x)=x$$ By the chain rule and product rule: $$\frac{dW}{dx}( e^x+xe^x ) =1$$ $$\frac{dW}{dx}=(e^x+xe^x)^{-1}$$ What is the error here? What is...
  2. 4

    Integrating Friedmann Equation of Multi-component universe respect to a and t

    I am having a trouble finding relationship between 'a' and 't' by integrating friedmann equation in a multi-component universe. It would be very helpful if you can help me with just matter-curvature only universe and matter-lambda only universe. The two integrals looks like following...
  3. PhizKid

    Maxima: Troubleshooting Integrating Issues

    Here is a screenshot: http://i.imgur.com/AG3EL.png Does anyone know what I am doing wrong here? I'm used to Mathematica because it's more syntax-lax but I guess in Maxima it's more explicit and strict. I've looked at examples online and they seem no different than what I have input, so I...
  4. F

    Integrating with a dirac delta function

    Homework Statement I have to integrate: \int_0^x \delta(x-y)f(y)dy Homework Equations The Attempt at a Solution I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity: \delta(0)=\infty I have to express the integral in terms of function...
  5. karush

    MHB Integrating exponential functions

    $\int x (5^{-x^2})dx= -\frac{1}{2} \int 5^{-x^{2}}(-2x)dx$ how is $\frac{1}{2}$ in front of the $\int$ derived
  6. D

    MHB Integrating delta/Bessel function

    \begin{alignat*}{3} u_t(r,\theta,0) & = & \delta(\mathbf{x} - \mathbf{x}_0) & = & \delta(r - r_0, \theta - \theta_0) \end{alignat*} $$ \int_A\delta(\mathbf{x} - \mathbf{x}_0)f(r,\theta)dA = \int_0^{2\pi}\int_0^a\delta(r - r_0, \theta - \theta_0)f(r,\theta)rdrd\theta = f(\mathbf{x}_0) $$ How do...
  7. D

    MHB Integrating Ln $(x)^p$: Step-by-Step Guide

    $$ \int (\ln x)^pdx, \quad p > 0 $$ How do I go about integrating this?
  8. P

    Integrating out of the real domain of a function

    Check this out: \int_{-1}^{0} ln(x) dx u=ln(x), dv=dx du=\frac{1}{x},v=x \int_{-1}^{0} ln(x) dx= x\space ln(x) \|_{-1}^{0} -\int_{-1}^{0} \frac{x}{x} dx =x\space ln(x) \|_{-1}^{0}-x \|_{-1}^{0} =\lim_{x\to0} x\space ln(x)-(-ln(-1))-(0-(-1)) \lim_{x\to0} x\space ln(x)=lim_{x\to0}...
  9. L

    Question about integrating w/ specific time

    When I integrate v(t) after I calculated the t, what do I get? Let;s say t=2. do I get the position vector in this 2s? If so it means the integral goes from 0 to t? or I still need to calculate from 2 to t? I could simply integrate v from 0 to t then place 2 in the position vector but I got an...
  10. N

    Linear equations - use of integrating factor

    Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great.Consider the equation \frac{dy}{dx} + 5y = e2x In this case, we begin multiplying both sides by e5x. This gives...
  11. N

    Linear equations - use of integrating factor

    Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great. Consider the equation \frac{dy}{dx} + 5y = e2x In this case, we begin multiplying both sides by e5x. This...
  12. D

    MHB What is the moral of conflicting definitions of the Heaviside function?

    Integrating the delta function: $$ \frac{4}{\pi^2}\int_0^{\pi}\int_0^{\pi}\delta(x - x_0,y - y_0)\sin nx\sin my dxdy $$ Would the solution be $\frac{4}{\pi^2}\sin nx_0\sin my_0$?
  13. F

    Calculating Distance Traveled Using Velocity Function and Integration

    Homework Statement if v = 8t - 3t^2 and the body is at P find where the body is from P when t=3 The Attempt at a Solution I am very new to calculus and have just been taught basics of differentiation and integration. I know that when you integrate there is an arbitrary constant so I have...
  14. V

    Fortran Interpolating & Integrating Data from File to 2D Array

    I have this code that i have been working on that will interpolate and integrate some data. I need to read this data from a file with unknown rows and columns to a two dimensional array/matrix so that the data can be broken down further into one dimensional arrays from the parent 2D array's...
  15. K

    How can I solve this without using integrating factor?

    Let the Wronskian between the functions f and g to be 3e^{4t}, if f(t) = e^{2t}, then what is g(t)? So the Wronskian setup is pretty easy W(t) = fg' - f'g = 3e^{4t} f = e^{2t} f' = 2e^{2t} So plugging it in I would get: e^{2t}g' - 2e^{2t}g = 3e^{4t} Which results in g' - 2g...
  16. O

    Integrating Factors for Stochastic Differential Equations

    Whenever I'm given a SDE problem that requires us to multiply both sides by an "integrating-factor", it's always given to us as a *Hint*. I would like to know how to come up with these integrating factors. Here's some examples: 1) For the mean-reverting Ornstein-Uhlenbeck (OU) SDE dX_t =...
  17. P

    Integrating hyperbolic functions

    Hi, I am trying to integrate (tanh(x)+coth(x))/((cosh(x))^2) I am substituting u=tanh(x), du=dx/((cosh(x))^2) and end up with 1/2(tanh(x))^2 + ln |tanh(x)| + C which is incorrect. What am I doing wrong??
  18. ElijahRockers

    Integrating absolute values over infinity

    Homework Statement Find <x> in terms of X0 if X0 is constant and \Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}} and <x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx where Psi* is the complex conjugate of Psi. Since there is no imaginary component, this is effectively Psi2. so, from...
  19. J

    Integrating 1/[1+sq.root(tanx)]

    Homework Statement How to integrate ∫1/(1+√tanx) dx? 2. The attempt at a solution Tried to substitute √tanx with z, but ultimately getting messed up with large expression
  20. B

    Mastering Integration by Parts: A Calculus 1 Student's Journey

    Hi all. I am having Calculus 1 this year. We are using a book called Thomas Calculus. I think its a lot of fun, but I have to work very much since there is basic stuff like trigonometry that I know really bad. Since I work so much with math I thought it could be fun and helpful to talk with...
  21. S

    Integrating $$\frac{ds}{(2y^2+1)^{3/2}}$$ on a Parabolic Curve

    Homework Statement Find $$\int_{C} \frac{ds}{(2y^2+1)^{3/2}}$$ where $$C$$ is the parabola $$ z^2=x^2+y^2 , x+z=1$$ Homework Equations The Attempt at a Solution I tried to parametrize the C , s.t$$ x=t, z=1-t, y=\sqrt{2t-1}$$ , but it seems to become a mess, and I don't know the...
  22. W

    Integrating Factor for First Order Linear Differential Equation

    Homework Statement Find an integrating factor for the first order linear differential equation \frac{dy}{dx} - \frac{y}{x} = xe^{2x} and hence find its general solution Homework Equations The Attempt at a Solution I found the integrating factor which is e^{-lnx} = x^{-1} and...
  23. A

    Integrating to determine speed as a function of time

    Homework Statement Homework Equations The above formulas The Attempt at a Solution I'm lost on where to start with this. The object has an intial velocity in the X direction and has the resistive force of the plontons acting upon it when it lands. What exactly is the equation...
  24. F

    Find Electric Potential by integrating the electric field.

    My question is: If you have a E= Ex+Ey+Ez To find V(x,y,z), i should: Just integrate Ex in order x? ∂V(x,y,z)/∂x = -Ex so: -V(x,y,z) = ∫Exdx or have i to sum the three integrals? -V(x,y,z) = ∫Exdx + ∫Eydy +∫Ezdz
  25. H

    Integrating F.dx: Constant μ, m & g Affect Force?

    I'm sure this is an easy question to answer - per the attached image, can someone please explain why the friction force has not required/been affected by integration? My calculus is very rusty but the only reason I can think of is that μ, m and g are all constants (for the purposes of the...
  26. B

    Integrating to an absolute value?

    Homework Statement I need some help understanding an integral step in the example below. I get how the integrand was set up, but I don't get how comes to two expressions with the absolute value of z-L/2. (Problem description if that is needed: L is the length of a cylinder with radius R, and P...
  27. B

    Finding Center of Mass of Nonuniform Rod: Integrating Over dm

    When looking at example 3 in this pdf: http://www.physics.isu.edu/~hackmart/centerofmass.pdf where it shows how to find the Center of Mass of a Nonuniform Rod. I was wondering when or what you have to know in order to integrate over dm rather than changing the variables in terms of dx.
  28. fluidistic

    First order DE with integrating factor

    Homework Statement I'm having a hard time to solve the following DE using an integrating factor, I'm asked to find one and solve the DE. (x+y^2)-2xyy'=0.Homework Equations The Attempt at a Solution If I call u=u(x) (I assume the integrating factor depends only on x, not on y), the following...
  29. C

    Limits and integrating over a singularity

    Suppose one needs to evaluate a definite integral over a singularity, like: -\int_{-1}^3 \frac{1}{x^2} dx The textbook way to do so is to split the integral into two parts around the singularity and take the limit, like so: \lim_{b\rightarrow 0} -\int_{-1}^b \frac{1}{x^2} dx and...
  30. C

    Integrating the Expectation Value

    I am asked a problem where I'm supposed to integrate the expectation value of a dynamic variable (operator) to solve a differential equation. OK, is the expectation value supposed to be a variable? But it seems to me like its a definite integral over allspace and thus is a number. So...
  31. J

    Integrating cot^4 x (csc^4 x) dx Using Identities and U Substitution

    ∫(cot^4 x) (csc^4 x) dx Wolfram wants to use the reduction formula, but I'm meant to do this just using identities and u substitution. I was thinking something along the lines of: =∫cot^4 x (cot^2 x + 1)^2 dx =∫cot^8 x + 2cot^6 x + cot^4 x dx but I don't know where to go from there.
  32. B

    Solving (x^2+y)dx + (x+e^x)dy = 0 with Integrating Factor

    Homework Statement From Elementary Differential Equation by Boyce and Diprima Chapter 2 Miscellaneous Problems #11 (x^2+y)dx + (x+e^x)dy = 0 ANS:(x^3/3)+xy+e^x=c Homework Equations multiplying an integrating factor to make the DE exact: 1. du/dx = u(My - Nx)/ N 2. du/dx = u(Nx-My)/ M The...
  33. L

    Integrating trig powers, why not by parts?

    Homework Statement Please help me understand the reason for substituting various trig identities into trig functions with powers instead of integration by parts. Does integration by parts not work on trig functions with powers, or is it just so much work that substituting trig identities to...
  34. C

    Understanding an integrating ADC

    http://www.hardwaresecrets.com/imageview.php?image=3740 Referencing the above image, I have a few a questions. 1) I think CLK means clock input; does CLR perhaps mean clear? 2) What exactly is the role of the output buffer? 3) I'm confused as to how the control logic exactly functions and...
  35. X

    How do I find the mass of a cone with changing density?

    Hi i was wondering if i could get help with a question. A cone has a radius of 0.75m and a height of 4m. The density changes throughout the cone and can be modeled by the function ρ=3h^2+4h (h for height). Find the mass of the cone. Please help
  36. N

    Integrating [itex]\frac{\partial f(x,y)}{\partial y(x)}[/itex] with respect to x

    I don't know how to apply Leibniz's rule for this integrand. I believe there is a substitution to allow me to express this integral differently, but the partial being a function of what we're integrating by is confusing me. \int^{b}_{a} \frac{\partial f(x,y)}{\partial y(x)} dx Knowns are that...
  37. T

    Integrating velocity with respect to time when velocity depends on position

    Okay this seems like it should be simple but I'm stuck. A particle moves from point A to point B in one dimension. The velocity of the particle changes with position according to some formula V = V(x). How long does it take for the particle to get from point A to point B? If I could write...
  38. A

    Integrating cot(x) by partswhat's wrong?

    Homework Statement I=∫cot(x)dx Homework Equations wht's wrong in this approach (integrating by parts) The Attempt at a Solution...
  39. T

    What is the charge density in a sphere with a given electric potential?

    Homework Statement Find the electric charge centred in a sphere of radius a, centered at the origin where the electric potential is found to be (in spherical coordinates) V(r)=kr^-2 where k is some constant. The Attempt at a Solution We have E=-\nabla V = -2kr^{-3} \hat{r} So...
  40. Jameson

    MHB Integrating Desmos Graphing Software with MHB

    We are in the process of talking with www.desmos.com about how we can integrate their graphing software on MHB in an easy-to-use way, similar to the Wolfram Widgets (hopefully). In the meantime I recommend their website for graphing 2-d equations. It's very easy to use and in my opinion is...
  41. N

    Mathematica Need Help Integrating w[x] with Mathematica

    Hi! I hope this is the right forum... I am having trouble with Mathematica: Maybe someone could help out of his experience ? I created w[x]:=RandomReal[NormalDistribution[0,1],x] I got it's plot. Now I want to integrate it: I wish to receive another graph, which is the plot of the...
  42. S

    Integrating √(1+x2)/x: A Step-by-Step Guide

    Homework Statement ∫√(1+x2)/x dx Homework Equations The Attempt at a Solution Let u = 1+x2 du = 2xdx 1/2du=xdx x=√(u-1) ∫√1+x2/x dx =∫√u/√(u-1) du or is it 1/2∫√udu as xdx would remove it. This is where I got confused.
  43. F

    In integrating to find a volume when is the function included?

    I just want to clarify this. I have had trouble before with including the function in the integral when I am trying to find volume. I have come to think that this is necessarily only when it is a double integral. But I had a specific question I posted earlier and I feel like somebody told me...
  44. T

    How can integration by substitution be used to solve complex equations?

    I have a maths B assignment where I need to find the volume of a tunnel. We have been given two equations to describe the area of this tunnel. The teacher says that we haven't learned how to integrate at this level, and he has also said that the chain rule and product rule do not work for...
  45. M

    Integrating a complicated polynomial

    Homework Statement Hi An integration question: t= 1/(1+r2) Can you please show me how to integrate with respect to r. Thank you. The Attempt at a Solution ∴t = (1+r2)-1 I then tried substituting u = 1+r^2 but that didn't work! Is there a trick with this?
  46. B

    MHB Integrating $\frac{1}{x^r-1}$ with r>1

    Hii Everyone, \int\frac{1}{x^{r}-1}dx where r is a real no. greater than 1regards, Bincy
  47. A

    Problem with Integrating by parts

    I would like to integrate by parts this term- \mu^2 (\nabla^2)^{-1} (\nabla\times B)\dot{B} Here B is a vector and \dot{B} is the time derivative of B. And \mu is just a constant. Can anyone help me?
  48. M

    Integrating Rational Functions

    1. ∫\frac{dx}{x3 + 2x} We're suppose to evaluate the integral. Use Partial Fraction Decomposition: \frac{1}{x3 + 2x} = \frac{A}{x} + \frac{Bx + C}{x2 + 2} 1 = A(x2 + 2) + (Bx + C)(x) 1 = Ax2 + 2A + Bx2 + Cx 1 = x2( A + B) + Cx + 2A Solving for A gives \frac{1}{2} Solving for B...
  49. E

    ODE Integrating factors found by inspection

    Homework Statement I'm reading a chapter out of Elem. DE 6th Edition by Rainville and Bedient (Ch 4 pg 61) titled integrating factors found by inspection. To explain it, the authors start with an equation, which is grouped to become: y dx + x dy + x^3y^2 dy = 0 which then becomes...
  50. P

    Integrating Web applications and Mainframe.

    Hi All, I have a very basic question on integrating two applications of different technologies. Let's take a web and a mainframe application, and we all know this could be integrated through 'web services'. But is it possible to integrate these two applications without using webservices...
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