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. C

    Finding torque by integration of weight

    Homework Statement Hi. I ve got a problem, where I have to show torque by integrating the weight of the rod over the whole it's length. Homework Equations [/B] Result, what I am suppose to get is: ## \tau_{rod} = \frac{mgb}2 ##The Attempt at a Solution [/B] When I try to integrate, I am...
  2. A

    Is e^(ikx)/(x^2+a^2) Integrateable over All Reals?

    Homework Statement (e^(ikx)/(x^2+a^2))dx (-infinity, +infinitiy) Isn't this function odd so it should be zero?Homework EquationsThe Attempt at a Solution I know how to complete the entire problem but I'm having troubles integrating this. I'm looking for someone to reference ( a website or an...
  3. L

    Confused about integrating acceleration to get velocity

    Homework Statement acceleratin as function of time ##a(t)= 2t+1## we know that v(0)=0 and s(0)=0 find v(t) find v(5) find s(t) find s(3) and I was thinking about also what happens when t is negative number, is it possible to find also v(-2)? what about s(-3)? Homework Equations integration...
  4. M

    Python Integrating to Infinity Numerically

    Hi PF! I am trying to integrate functions over an infinite domain. One example is $$\int_0^\infty \frac{e^{-x}}{\sqrt{x}}\,dx$$ I know the substitution ##u = \sqrt{x}## reduces this problem to integrating ##\exp(-x^2)##, but if I want to integrate the function as is, how would I do this? I've...
  5. Saracen Rue

    B Solving Integral Equation: $\frac{\sin\left(x\right)}{\cos^{3}\left(x\right)}$

    Hi all, just a quick question: I'm trying to integrate this function in two different ways and I'm getting a different answer each way, can someone please quickly tell me where I'm going wrong? I've read through it for a couple hours and can't pick up the mistake. ##\int _{ }^{...
  6. B

    I Multiplying by dr/dt and integrating with respect to t

    I came across this step in a derivation:$$m\ddot{r}=\frac{L^2}{mr^3} -V'(r)$$ Multiplying by ##\dot{r}## and integrating with respect to t to get $$\frac{1}{2}m\dot{r}^2+\frac{L^2}{2mr^2}+V(r) = C$$ I am not very clear about how the 1st term came to this. Can some one gives a pointer?
  7. Drakkith

    Can you find the integrating factor for this non-exact equation?

    Homework Statement Find an integrating factor of the form ##x^Ay^B## and solve the equation. ##(2y^2-6xy)dx+(3xy-4x^2)dy=0## Homework Equations ##M=2y^2-6xy## ##N=3xy-4x^2## ##IF = exp(\int \frac{M_y-N_x}{N}\,dx)## or ##IF = exp(\int \frac{N_x-M_y}{M}\,dy)## The Attempt at a Solution [/B]...
  8. Saracen Rue

    B Integrating to infinity issue

    Say we have two functions; ##f\left(x\right)=\frac{1}{e^x-1}## and ##g\left(x\right)=\ln \left(\frac{1}{x}+1\right)##. Let us find the limit of both functions as x approaches infinity; ##\lim_{x \rightarrow \infty} {f(x)} = \frac{1}{e^\infty-1} = \frac{1}{\infty} = 0## Therefore as ##x...
  9. J

    2D Integrating With Quadratic Arg. of Delta Function

    Homework Statement I have a 2D integral that contains a delta function: ##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##, where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
  10. U

    B Integrating to find surface area/volume of hemisphere

    To find the surface area of a hemisphere of radius ##R##, we can do so by summing up rings of height ##Rd\theta## (arc length) and radius ##r=Rcos(\theta)##. So the surface area is then ##S=\int_0^{\frac{\pi}{2}}2\pi (Rcos(\theta))Rd\theta=2\pi R^2\int_0^{\frac{\pi}{2}}cos(\theta)d\theta=2\pi...
  11. Brian

    I Integrating function of force with respect to angle

    I have a function of force with respect to a certain angle, θ. I also have the angle of the force, Ψ, with respect to the same angle. Plug in θ, get the force and Ψ. This function is determined by a table of data: F(θ) and Ψ(θ) This force is acted upon a beam at a distance L from a pivot at one...
  12. R

    I Integrating a curve of position vectors

    I'm looking at different ways to express the derivative a curve, like circular and tangent/normal components. Is there no such way that let's you express a vector integral in terms of information from the vector you want to integrate?
  13. H

    I Integrating Equation 1: Understanding the Answer

    I am working my way through a textbook, and whenever this equation is solved (integrated), the answer is given as: u = f(x) + f(y) I don't understand it. If I integrate it once (with respect to y, say), then I obtain: ∂u/∂x = f(x) -----eq.1 If I integrate again (this time with respect to...
  14. binbagsss

    Is There an Integration Factor for This Differential Equation?

    Homework Statement ##cos t \frac{dv}{dt} + (sin t) t = \frac{GM}{b^2 }\sin^3 t ## Homework Equations above The Attempt at a Solution im pretty stuck to be honest. It almost looks like a product rule on the LHS but it has the wrong sign, RHS I've tried writing ##sin^3 t## as ##(1-cos^2t)\sin...
  15. R

    I Can a Velocity Vector be Integrated to Parameterize y=x^2?

    Hi (1/sqrt(4t²+1), 2t/sqrt(4t²+1)) gives a unit tangent to the curve y=x^2 at point (t,t^2). Viewing the vector as velocity, shouldn't I be able to integrate it and get a parameterization for y=x^2?
  16. T

    I Integrating both sides of an equation question

    I am studying a paper and a math step like this was used: dt'=(1+\frac{h}{2}sin^2(\theta))dr \\ \int^{t1}_t dt'=\int^d_0 (1+\frac{h}{2}sin^2(\theta))dr \\ where\\ h=h(t-\frac{r}{c}-\frac{r}{c}cos(\theta)) This seems wrong because it seems to me that you're not doing the same thing to both...
  17. H

    I How to find the integrating factor? (1st order ODE)

    x2 + y + y2dx - x dy = 0 Integrating factor, I(x,y) = -1 / (x2 + y2) How to find the integrating factor ?Why I cannot use below method to solve the ode ? (1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR (1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)
  18. T

    I Integrating the Friedmann Equations

    Hello, I was just wondering about the method for integrating Friedmann equation for k=1 and w=1/3 ∫(8πρ/a^(2)-1)^-1/2thank you
  19. karush

    MHB Integrating $\sec^2(2x)$: A Puzzler

    \begin{align*}\displaystyle I_{7}&=\int_{0}^{\pi/8}\frac{\sec^2(2x)}{2+\tan\left({2x}\right)} \\ &= \end{align*} not sure of the u substitution here... if $u=tan(2x)$ then $du=2sec^2(2x)$ but stuck with 2 in the denominator after subst
  20. T

    Integrating with respect to area? Past paper question

    This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me). The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...
  21. Mr Davis 97

    I Integrating Sec^3(x) without Absolute Value

    So I am trying compute ##\displaystyle \int \sqrt{1+x^2}dx##. To start, I make the substitution ##u=\tan x##. After manipulation, this gives us ##\displaystyle \int |\sec u| \sec^2u ~du##. How do I get rid of the absolute value sign, so that I can go about integrating ##\sec^3 u##? Is there an...
  22. P

    I Integrating sqrt(x) cos(sqrt(x)) dx

    Question: sqrt(x) cos(sqrt(x)) dx My try: Let dv = cos(√x) => v = 2√xsin(√x) and u = √x => du = dx/(2√x) Using integration by parts, we get ∫√x cos(√x) dx = 2√x√x sin(√x) - ∫(2√xsin(√x) dx)/(2√x) = 2x sin(√x) - ∫sin(√x) dx = 2x sin(√x) + 2 cos(√x) √x However, the answer given in the book...
  23. J

    B Elementary question on integrating an equation

    It is a very simple question. If we have an expression like this one: x + y = 2 And we have to differenciate it, there is an algorithm that tells us how to do it. We have to find the relationship between the differentials of the given functions. To find them we have to substract the...
  24. O

    I Integrating x-squared

    OK, I admit: this will be the most idiotic question I have ever asked (maybe: there could be more) So, I am aware of the differential calculus (derivatives) and the integral calculus (integrals). And separate from that, there is the first fundamental theorem (FFT) of the calculus which relates...
  25. karush

    MHB -206.q3.2 method of integrating factor

    2000 $\tiny{206.q3.2}\\$ $\textsf{3. use the method of integrating factor}\\$ $\textsf{to find the general solution to the first order linear differential equation}\\$ \begin{align} \displaystyle \frac{dy}{dx}+5y=10x \end{align} $\textit{clueless !}$
  26. Kara386

    Integrating Maxwell Boltzmann Distribution in One Dimension

    Homework Statement I need to integrate ##\langle |v_x| \rangle = \int^{\infty}_0 |v_x| \sqrt{\frac{m}{2\pi kT}}e^{-mv_x^2/2kT}dv## For context this is a Maxwell Boltzmann distribution in one dimension, and I've actually been asked to calculate ##\langle v_x \rangle## which is given by...
  27. D

    I Integrating imaginary units and operators

    When integrating terms including the imaginary unit i and operators like position and momentum, do you simply treat these as constants?
  28. C

    I Find potential integrating on segments parallel to axes

    A simple method to find the potential of a conservative vector field defined on a domain ##D## is to calculate the integral $$U(x,y,z)=\int_{\gamma} F \cdot ds$$ On a curve ##\gamma## that is made of segments parallel to the coordinate axes, that start from a chosen point ##(x_0,y_0,z_0)##. I...
  29. dykuma

    Integrating Complex Functions in the Complex Plane

    Homework Statement Evaluate the following line integrals in the complex plane by direct integration. Homework Equations Z= x+i y = Cos(θ) +i Sin(θ) = e^i*θ The Attempt at a Solution I'm not sure how to evaluated this by hand. I tried using Z= x+i y = Cos(θ) +i Sin(θ), and evaluating the...
  30. toforfiltum

    Integrating triple integral over region W

    Homework Statement $$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##. Homework EquationsThe Attempt at a Solution Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have...
  31. NihalRi

    Integrating Functions with Only One Variable: A Guide for Beginners

    1. Homework Statement I'm trying to integrate this, the only variable is y the others(x,w) are all constants. Homework Equations The ways of integrating that I am familiar with are substitution, trigonometric substitution, by parts & partial fraction decomposition. The Attempt at a Solution...
  32. G

    How did they get 1=A^2(L/4) when integrating?

    I was looking for questions to practice normalizing a wave function, so I visited the following online pdf, http://people.physics.tamu.edu/syeager/teaching/222/hw1solution.pdf. The first question was to find the normalization constant, A of ψ(x) = A cos (2πx/L) for (−L/4) ≤ x ≤ (L/4). After...
  33. B

    A Integrating definite Heaviside function

    I am trying to integrate the following triple integral, which has a heaviside function in the inner most integral:$$ \frac{16}{c_{4}^{4}} \int_{0}^{c_{4}} c_{3}dc_{3} \int_{c_{3}}^{c_{4}} \frac{dc_{2}}{c_{2}} \int_{0}^{c_{2}}f(x)\left ( 1-H\left ( x-\left ( c_{4}-a \right ) \right ) \right )dx...
  34. nysnacc

    Rate of change and integrating factor

    Homework Statement Initially 5 grams of salt are dissolved in 20 liters of water. Brine with concentration of salt 2 grams of salt per lter is added at a rate of 3 liters per minute. The tank is mixed well and is drained at 3 liters a minute. How long does the process have to continue until...
  35. E

    I Integrating factor of (y+1)dx+(4x-y)dy=0

    I tried to put it in standard form as (dx/dy)+4x(1/(1+y))=y/(y+1). I get that the integrating factor is (y+1) but i am not sure if i am doing it right or what am I suppose to do next? I get (y+1) because the integrating of 1/(y+1) is ln(y+1) and since it has e, then ln cancels and i am left...
  36. M

    MHB Integrating e^X - e^-x/e^-x+1 dx

    how do you integrate e^X-e^-x/e^-x+1 dx i am trying multiplying by e^x and trying to make it into no fraction but i am having no luck
  37. StudentOfScience

    Studying How to Study Science: Integrating Spaced Repetition Review

    Note: for easy reference, "serious" questions and such are italicized. I also should note that although the title says "science", I include many references to mathematics (I suppose one could argue that it is a science in some senses). Furthermore, in another thread that I've posted, some people...
  38. P

    I Is the author integrating constants?

    Dear Sirs, I am currently calculating a velocity profile of an annular flow. Unfortunatelly I am not understanding the following step: [PLAIN]http://[url=https://postimg.org/image/vl256ffhj/] That seems the author had integrated the R constant. And remains the question: why had R been...
  39. T

    MHB Integrating $\int_{1}^{\infty} {x}^{-3} lnx \,dx$: Is the Answer 1?

    I have $$\int_{1}^{\infty} {x}^{-3} lnx \,dx$$ I choose to use integration by parts so I let $u = lnx$ and $dv = {x}^{-3} dx$. Therefore $du = \frac{1}{x} dx$ and $v = \frac{{x}^{-2}}{-2}$ Thus, what I need to evaluate is $$- \frac{-lnx}{{2x}^{2}} - \frac{1}{4 {x}^{2}} + C$$ As $x$...
  40. chwala

    Solving a differential equation using integrating factor

    Homework Statement Solve ##{dy/dx}-2xy=2x##Homework EquationsThe Attempt at a Solution Let ##P= -2x ## and Q= 2x, Integrating factor =## e^{-x^2} ## ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## ##y.e^{-x^2}={x^2} e^{-x^2}+∫ 2{x^3} e^{-x^2}dx## since ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## then...
  41. karush

    MHB Integrating by Parts: Solving a Sin x Problem

    \\text{w8.4.13 Integration by Parts} nmh{2000} $\displaystyle I=\int \sin\left({\sqrt{x}}\right) \ d{t} =2\sin\left({\sqrt{x}}\right) -2\sqrt{x}\cos\left({\sqrt{x}}\right)+C$ $\begin{align} \displaystyle u& = {\sin\left({\sqrt{x}}\right)} & dv&={1} \ dx \\ \\...
  42. C

    Mathematica Trying to define a integrating function in Mathematica

    I need to define a function that integrates a function in some interval and returns its numerical value. I am not allowed to use built in functions for integrating in Mathematica. This is my code that won't work: http://pokit.org/get/?93856fe8f2070ba781028f8634b9ac3a.jpg This is a code from a...
  43. D

    I Integrating Force: Is F=(m/t)*v Correct?

    If force isn't constant and you want to find the force of something at a specific time, you'd integrate right? For a specific question I came up with this F=ma Integral (m * da) da=dv/t Integral m * dv/t (m/t) * dv (m/t) * v So F = (m/t) * v Does this seem right? Or I'm I making illegal...
  44. Jatin Kaushal

    B Why do we put the dot multiplication symbol when integrating

    This is I think a really dumb question, but I never got it, why do we have that dot symbol when we integrate. Like in gauss's law, we have ∫E⋅dA . why is that ⋅ there? Thank you for your help
  45. evinda

    MHB Integrating by an other variable

    Hello! (Wave) I am looking at a proof where we have: $$\int_{-\infty}^{+\infty} |u|^{\frac{n}{n-1}} dx_1 \leq \left( \int_{-\infty}^{+\infty} |Du| dy_1 \right)^{\frac{1}{n-1}} \left( \prod_{i=2}^n \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} |Du| dx_1 dy_i\right)^{\frac{1}{n-1}}$$...
  46. R

    Integrating Sums (Laplace Transform)

    Homework Statement Using: \mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0} Give a formula for the Laplace transform of an arbitrary nth degree polynomial p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n Homework Equations \mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt The...
  47. J

    Integrating the solar wind equation

    Hi, eveyone have been struggling to do this problem for a long time now, figured it is something very simple I am missing so thought I should ask here. 1. Homework Statement Parker's solar wind equation is given after some manipulation as: (v - (Cs2 / v) dv/dr = 2 (Cs2 / r2) (r - rc ) where...
  48. O

    A Integrating the topics of forms, manifolds, and algebra

    Hello, As you might discern from previous posts, I have been teaching myself: Calculus on manifolds Differential forms Lie Algebra, Group Push forward, pull back. I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...
  49. S

    B Integrating velocity equation problem

    I've already completed the first question, but with number two, it's a different case. Here's my attempt: \frac { d{ v }_{ y } }{ dt } \quad =\quad -g\quad -\quad \beta { v }_{ y }\\ \frac { d{ v }_{ y } }{ -g\quad -\quad \beta { v }_{ y } } \quad =\quad dt\\ \int { \frac { d{ v }_{ y } }{...
  50. Bounceback

    B Integrating any e^f(x) function

    Solve f'(x)*a(x)+a'(x)=1 For a(x) (there's another less important question at the bottom) Background behind equation (trying to find a function to integrate any e^f(x)): \int e^{f(x)}\,dx=e^{f(x)}*a(x) e^{f(x)}=e^{f(x)}*{f'(x)}*a(x)+e^{f(x)}*a'(x) 1=f'(x)*a(x)+a'(x)A few of my attempts...
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