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. btouellette

    Integrating monochromatic luminosity to get total luminosity

    Homework Statement Integrate the monochromatic luminosity (of a blackbody model star) over all wavelengths to obtain an expression for the total luminosity. This is 3.14(a) from An Introduction to Modern Astrophysics 2e by Carroll and Ostlie. Homework Equations ## L_\lambda d\lambda =...
  2. L

    Integrating to get r^2 = l^2 + a^2

    I'm reading a physics book( "Einstein's gravity in a nutshell" Zee) and at page 126 the author says: (dr/dl)^2 + a^2 / r^2 = 1 gives r^2 = l^2 + a^2 where we absorbed an integration constant into l by setting l=0 when r=a. Can someone explain what's going on here? How do I have to "massage"...
  3. L

    Integrating F over a Paraboloid Region

    Homework Statement Let F = <x, z, xz> evaluate ∫∫F⋅dS for the following region: x2+y2≤z≤1 and x≥0 Homework Equations Gauss Theorem ∫∫∫(∇⋅F)dV = ∫∫F⋅dS The Attempt at a Solution This is the graph of the entire function: Thank you Wolfram Alpha. But my surface is just the half of this...
  4. O

    Integrating pressure over area to get friction force

    I'm doing some experiments where I need to calculate the resistance force on a cylindrical body (cable) when it's being pulled through water saturated sand We derived formula from a theory which was originally based on a square body by using stress components. This way we know the pressure at...
  5. B

    Integrating over the unit circle

    Homework Statement Suppose that the function ##f(z)## is analytic and that ##|f(z)| \le 1## for all ##|z| = 1##. Homework EquationsThe Attempt at a Solution I was hoping someone could verify my work. Okay, if I understand correctly, ##|f(z)| \le 1## is true for all all complex numbers ##z##...
  6. R

    Quick question about integrating limits in QM problems

    Am I correct in assuming that if there is a potential present and it is not infinite then integrals will always be made from minus infinity to infinity, but where an infinite potential exists then the integral will depend on the size of the confinement area? Sorry to be a little disambiguous...
  7. B

    Integrating a Complex Function Over a Contour

    Homework Statement ##z(t) = t + it^2## and ##f(z) = z^2 = (x^2 - y^2) + 2iyx## Homework EquationsThe Attempt at a Solution Because ##f(z)## is analytic everywhere in the plane, the integral of ##f(z)## between the points ##z(1) = (1,1)## and ##z(3) = (3,9)## is independent of the contour (the...
  8. F

    Problem integrating complex function

    Homework Statement Hello, I have been tasked with the next problem, I have to prove that the next two integrals are complex numbers; but I have no idea of how to attack this problem. Homework Equations ∫dx f*(x) x (-ih) (∂/∂x) f(x) integrating between -∞ and ∞ ∫dx f*(x) (-ih) (∂/∂x) (x f(x))...
  9. TheFerruccio

    Integrating until symmetric bilinear form

    Homework Statement I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here. Suppose I have a square domain over which I am integrating two functions w and...
  10. I

    MHB Integrating y^3 over Triangle D

    $\int \, \int_{D}^{} \, y^3 dA$ D is the triangular region with vertices (0,1), (1,2), (4,1) i can't get past this problem. i drew the triangle but i don't know how to find the intervals... excuse my ugly drawing :p
  11. gfd43tg

    Integrating Factor Homework: Solving Diff. Eq w/ Constants

    Homework Statement ##C_{B}## is a function of ##\tau'##, and ##k_{1}##,##k_{2}##, and ##C_{A0}## are constants. I want to solve this differential equation \frac {dC_{B}}{d \tau'} + k_{2}C_{B} = k_{1}C_{A0}e^{-k_{1} \tau'} Homework EquationsThe Attempt at a Solution Using the integrating...
  12. T

    Integrating from - to + infinity

    Homework Statement I am having trouble integrating ∫ (x = -∞ to +∞) x3e-αx2 dx part--is this 0 or 1/α2? And, could someone explain? I am pretty sure that, when ∫ (x =0 to +∞) x3e-αx2 dx = 1/α2 However, with x = -∞ to +∞, and the function of the equation being odd, I am lost.Homework Equations...
  13. T

    Solve sine^x Variation of Parameters: y"+3y'+2y

    Homework Statement Solve by variation of parameters: y" + 3y' + 2y = sinex Homework Equations Finding the complimentary yields: yc = c1e-x + c2e-2x The Attempt at a Solution I set up the Wronskians and got: μ1 = ∫e-2xsin(ex)dx μ2 = -∫e-xsin(ex)dx The problem is that I have no idea how to...
  14. B

    Integrating Radial Probability Densities: A Guided Explanation

    Homework Statement Image: http://puu.sh/ca93V/7eb9abf342.png Homework Equations Ok I know to use this guy http://puu.sh/ca95c/ad0155a4d6.png Which then turns into this http://puu.sh/ca96W/68802e045c.png (except from .5a to 4a, not 0 to a)The Attempt at a Solution I get lost on trying...
  15. C

    Integrating a delta function with a spherical volume integral

    Homework Statement Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity. Homework EquationsThe Attempt at a Solution This integral actually came up in a homework problem for my E&M class and I'm...
  16. cqfd

    Integrating sin(x)*cos(x) paradox

    I am encountering a paradox when calculating the integral ##\int sin(x)\cos(x)\,dx## with integration by parts: Defining ##u = sin(x), v' = cos(x)##: ##\int sin(x)cos(x) dx = sin^2(x) - \int cos(x) sin(x) dx## ##\Leftrightarrow \int sin(x) * cos(x) dx = +1/2*sin^2(x)##. On the other hand...
  17. T

    Integrating ma+kx=0 to get x(t)

    Homework Statement As stated in the title, I'm having trouble integrating ma+kx=0 to get x(t) Homework EquationsThe Attempt at a Solution So I know I have to integrate twice but I'm not getting the answer required. ∫a = -k/m∫x v = (-k/m)[(x²/2) + C] ∫v = (-k/2m)∫x² + (-kC/m) x =...
  18. T

    Integrating challenge I am having

    Hi, I am doing an exercise practice samples for the upcoming quiz, and stumbled across two questions I'm having trouble solving... First question is to integrate integral e-x2 dx ...where the solution is equal to pi1/2 Also... As for the second question (of a different equation) how can one...
  19. C

    Integrating ##d\psi=(x+y)dx +x_0dy##

    I am quite embarrassed to ask this question, as I know i have lost track of the concept here, but Ill nevertheless ask it. I was going through Mathematical methods for physicists (pg 333), and there was an example: "Solve $$y'+(1+\frac{y}{x}) = 0$$" My problem is, (a) when you put the...
  20. G

    Mathematica Can You Integrate a Function of Two Variables in Mathematica?

    I have a function of two variables F[x_,y_] and I Would like to integrate over one variable only and get a function G[x] for example and work with it. I want something like: G[x_]:=NIntegrate[F[x,y],{y,0,\infty}] But it doesn't work.
  21. C

    Integrating (x^4+x^2)^0.5 form 3^0.5 to -3^0.5

    The Attempt at a Solution Let x=tan u dx=sec^2(u)*du When x=3^0.5,u=pi/3 x=-3^0.5,u=-pi/3 S sec^2(u)d(sec u) =1/3[2^3-2^3]=0
  22. M

    Integrating 8/(xln(3x))dx | Solving for ln(3x) | Homework Help

    Homework Statement Integrate: (8)/(xln(3x))dx Homework Equations The Attempt at a Solution I separated the equations into 8/x and 1/(ln3x). I sub u for ln(3x) and I got 1/x for du. Since I had 8/x, I made it 8du. So the new integration will be 8/udu. I integrated so it will be...
  23. S

    Integrating sine where argument goes to infinity.

    After some integration, i am getting a form e^{i \alpha\phi+i\beta\phi\sin(\phi-\phi')-i\gamma\sin\phi} , where ##\alpha, \beta, \gamma## are constants. Now i want to apply the limit where ##\phi ## ranges from 0 to ##\infty ## (ya, in the argument of sine we will encounter ##\infty ## which is...
  24. M

    Integrating Subtraction?

    Let http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmex10/alpha/144/char5A.png f(x)dx=5 a=7, b= 13http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmmi10/alpha/144/char3B.png [PLAIN]http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmex10/alpha/144/char5A.png...
  25. _N3WTON_

    Integrating Factor Method: Finding the Solution to a Cosine Equation

    Homework Statement Find the general solution to the indicated equation: cos(x)y' + ysin(x) = 1 Homework Equations e^\int p(x)\,dx * y(x) = int\ f(x)\, dx e^\intp(x)\,dx + C The Attempt at a Solution Ok, I am having trouble getting started with this problem because I am not...
  26. M

    Integrating question but withe error

    Homework Statement Calculate the integral: ∫2x/(x2−11x+30) dx 2. The attempt at a solution I factored and got A/(x-6) + B/(x-5) = 2x/(x2−11x+30) Then I isolated and found A = 12 and B = -10 Then, after setting up the integral again, got 12ln(x-6) - 10ln(x-5) + C Unfortunately this is not...
  27. D

    Where did this 1/4*ln(C) term come from when integrating?

    Homework Statement The givenequation is this: \frac{1}{4} \frac {du}{(2-u)} \ + \ \frac{1}{4} \frac{du}{(2+u)} \ = \frac{dx}{x} My book says that when integrated, the above equation becomes \frac{-1}{4} \ln (2-u) \ + \ \frac{1}{4} \ln (2+u) \ = \ln (x) + \frac{1}{4} \ln (c) I...
  28. F

    When Integrating (2x)/(4x^(2)+2) I get two different integrals ?

    Hi So let's have ∫(2x)/(4x^(2)+2) dx Without factorising the 2 from the denominator, I integrate and I get 1/4*ln(4x^(2)+2)+c which makes sense as when I differentiate it I get the original derivative. BUT when I factor the 2 from the denominator I have 2x/[2(2x^(2)+1)]...
  29. T

    Integrating exponent to get delta function

    Something i ran into while doing hw Homework Statement starting with \int{dx} e^{-ikx}\delta(x) = 1 we conclude by Fourier theory that \int{dk} e^{+ikx} = \delta(x) Now, i try to compute \int{dk} e^{-ikx} (I've dropped the normalization factors of 2\pi. I believe no harm is done by...
  30. Mogarrr

    Integrating functions with absolute values

    To find E |X| of a cauchy random variable, I need to integrate \int_{-\infty}^{\infty}\frac1{\pi}\frac{|x|}{1+x^2}dx . From the definition of absolute value, we have \int_{-\infty}^0\frac1{\pi}\frac{-x}{1+x^2}dx + \int_0^{\infty}\frac1{\pi}\frac{x}{1+x^2}dx (I think). But, the very next...
  31. A

    How did we get to the constant in r\frac{\partial p}{\partial r}=c_1?

    Hi, When we have \frac{\partial}{\partial r}(r\frac{\partial p}{\partial r})=0 and we get r\frac{\partial p}{\partial r}=c_1 To get there, did we do this \int\frac{\partial}{\partial r}(r\frac{\partial p}{\partial r}) dr=\int 0 dr or \partial (r\frac{\partial p}{\partial r})=0\partial r...
  32. J

    Integrating Along C: Solving ∫ tan(z/2)/(z+π/2)(z-π/2)² dz

    Homework Statement ∫\frac{tan(\frac{z}{2})}{(z+\frac{\pi}{2})(z-\frac{\pi}{2})^{2}} dz integration along C: abs(z) = 4 (along the circle of radius is 4) Homework Equations Cauchy Integral FormulaThe Attempt at a Solution I tried to set g(z) that is analytic inside C but I cannt set it...
  33. Runei

    Integrating a differential (Problem in thermodynamic derivation)

    Hi, I'm looking at a derivation of the thermodynamics of black-body radiation. My question is in regards to the math of the derivation. Using the first law of thermodynamics and considering an adiabatic expansion of the cavity, it can be stated that dU = -\frac{u}{3}dV Here small u...
  34. M

    Using u substitution for integrating.

    So I am pretty bad at u substitution. I don't really get how to replace values with du or u. Can you please give me tips on how to do u substitution well? Thanks.
  35. K

    How do you integrate √(sinθ + 1)?

    I have a polar arc length problem that comes down to integrating √(sinθ + 1). Through double u-sub and trig sub I got it to be -2√(1 - sinθ) but that seems to be wrong. Wolfram Alpha says that the integral is [2√(sinθ + 1)(sin(θ/2) - cos(θ/2)] / [sin(θ/2) + cos(θ/2). I'm wondering how this is...
  36. G

    Integrating (1+cos(x))/sin(x) with Multiple Choices

    Homework Statement ∫(1+cos(x))/sin(x) dx This is a multiple choice with the following options a. Ln|1-cos(x)| +C b. Ln|1+cos(x)| +C c. sin(x) +C d. csc(x)+tan(x) + C e. csc(x) +cot(x) +C Homework Equations The Attempt at a Solution ∫(1+cos(x))/sinx dx )...
  37. S

    Proving Integral of Arcsech x using Integration by Parts

    Homework Statement I was asked to prove the integral ##\int_{\frac{4}{5}}^{1} \textrm{arcsech}(x) =2\arctan 2-\frac{\pi}{2}-\frac{4}{5} \ln 2##Homework Equations Integration by partsThe Attempt at a Solution Let ##u=\textrm{arcsech} (x)## ##\textrm{sech u}=x## ##\textrm{cosh u}=\frac{1}{x}##...
  38. G

    Integrating for approximation of a sum

    Homework Statement Find an N so that ##∑^{\infty}_{n=1}\frac{log(n)}{n^2}## is between ##∑^{N}_{n=1}\frac{log (n)}{n^2}## and ##∑^{N}_{n=1}\frac{log(n)}{n^2}+0.005.## Homework Equations Definite integration The Attempt at a Solution I began by taking a definite integral...
  39. B

    Integrating centrifugal force

    Homework Statement This is not homework. Suppose I am rotating a ball on a string (m=1, r = .2, v = 10 m/s) Homework Equations Fc = (m) v2/r = 500 N*m If I reduce the length of the string to .1, v becomes 20, so Fc = 4000 N*m what is the work I have done, what kind of integration do...
  40. B

    Integrating Infinities: Zero or Infinity?

    Hi! I have a question about integrating a function with an infinite value. If you integrate a function with a place where the integrand diverges to infinity, I understand that the value of the integral should diverge to infinity. However, what happens when you set both bounds to be the value...
  41. MarkFL

    MHB Integrate sqrt(1-x^4)/x^5 dx Using Trig Sub | Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  42. J

    Help integrating 1/(cosh(z)+1)

    Homework Statement integrate 1/(cosh(z)+1) Homework Equations The Attempt at a Solution integral(1/(cosh(z)+1))=arctan((cosh(z)) but can I also do 1/(cosh(z)+1)=cosh(z)-1/sinh(z) and go from there and get a simpler solution or something?
  43. xwolfhunter

    Representing three-dimensional shapes with functions and integrating

    So, I don't like calculus class, since it's very boring, but I do like math, and I intend to sort of become a mathematical autodidact. So I just thought I'd try to come up with a solution to a problem I created, and this was to integrate one of those gigantor Reese's easter egg things (holy...
  44. M

    Integrating dx/(2+sin(x)) using a complex substitution

    Homework Statement Compute the real integral \int\frac{dθ}{2+sin(θ)}, where the limits of integration are from 0 to 2π by writing the sine function in terms of the exponential function and making the substitution z=e^{iθ} to turn the real integral into a complex integral. Homework...
  45. F

    MHB Integrating $\frac{x^2}{(1+x^2)^3}$ Over the Real Line

    integrate $\frac{x^2}{(1+x^2)^3}$ over the real line
  46. J

    Solve Homogeneous D.E. integrating

    Homework Statement Dy/Dx = (Y-x)/(Y+x) Homework Equations Y=ux dy=udx+xdu The Attempt at a Solution Dy/Dx = (Y-x)/(Y+x) Plug in my substitutions udx+xdu(1/dx)=(ux/ux+x) - X/(ux+x) Simplify u+x(du/dx)=(ux)/x(u+1) - (x)/((x)(u+1)) u+x(du/dx)=u/(u+1) -(1)/(u+1)) u+x(du/dx)=u-1/(u+1)...
  47. G

    Exact ODE and Finding Integrating Factors

    Homework Statement In my ODE class, we learned how to solve first order ordinary differential equations which are not exact yet but exact after multiplying by the right integrating factor. The integrating factor we learned about take one of the five forms: f(x), f(y), f(xy), f(x/y), and...
  48. Y

    Integrating Kinematics for Velocity from Acceleration: A Simplified Approach

    When you want to get velocity from accelleration i have been told you integrate. Howver v=at and so surley you can just multiply each term in the accelleratin expression by t. ie: a=4-0.2t Surley you can just: v=(4-0.2t)t v=4t-0.2t2
  49. S

    Can Sin(pi*x^3) Be Integrated Using Elementary Functions?

    Homework Statement Need to integrate sin(pi*x^3) Got to the end of a long question and this is the final step but I can't seem do it! Homework Equations The Attempt at a Solution Tried substitution of u = x^3 and said dx = 1/3x^2 du but this doesn't cancel any x variable...
  50. Y

    MHB Integrating Complex Integrals: Exploring the Mystery of a "-1

    Hello, I have this integral here: \[\int e^{\sqrt{x}}dx\] and I wanted to ask, why can't I treat it like I would treat this integral: \[\int (3x+5)^{5}dx\] In which I would integrate as if g(x)=3x+5 is a normal x, and then divide by the inner derivative ? I tried it with the upper integral...
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