What is Integration: Definition and 1000 Discussions

System integration is defined in engineering as the process of bringing together the component sub-systems into one system (an aggregation of subsystems cooperating so that the system is able to deliver the overarching functionality) and ensuring that the subsystems function together as a system, and in information technology as the process of linking together different computing systems and software applications physically or functionally, to act as a coordinated whole.
The system integrator integrates discrete systems utilizing a variety of techniques such as computer networking, enterprise application integration, business process management or manual programming.System integration involves integrating existing, often disparate systems in such a way "that focuses on increasing value to the customer" (e.g., improved product quality and performance) while at the same time providing value to the company (e.g., reducing operational costs and improving response time). In the modern world connected by Internet, the role of system integration engineers is important: more and more systems are designed to connect, both within the system under construction and to systems that are already deployed.

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  1. C

    B Question about the limits of integration in a change of variables

    Hello everyone, If I have an integral ##\int_0^r \sqrt{(r^2 - x^2)}dx## and I'm integrating across the first quadrant to get the area of the first quater of a circle. And I change variables with ##x = r\cos{\theta}## and ##dx = -{r}\sin{\theta}{d\theta}## And I form a new integral that's...
  2. F

    Is an API Always Necessary for Server-Client Communication?

    It is clear that a server and a client are programs communicative iwth each other using one or more protocols (HTTP, TCP, etc.) I conceptually understand what an API is: it is like an intermediary between two programs that makes integration easy. For example, we build app A and want to connect...
  3. Y

    Mathematica table integration error

    I am calculating the temperature distribution and utilizing the obtained results to calculate the current distribution. In order to do this , I employ a table in which I stock all the current distribution for each value of radius . Subsequently, I aim to identify the radius corresponding to a...
  4. M

    I Help please integrating this function over a rectangular area

    Hi I struggle with integration generally. Could you be able please to talk me through the stages of this one? thanks martyn
  5. A

    Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

    I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ Which leads...
  6. J

    I Integration with different infinitesimal intervals

    Some sources state a similar format of the following $$\int_a^{a+da}f(x)dx=f(a)da$$ Which had me thinking whether the following integration can exist $$\int_a^{a+dx}f(x)dx=f(a)dx$$ I have difficulty grasping some aspects about these integrations 1. Regarding the 1st integration, shouldn't ##a##...
  7. chwala

    Integration of functions of form ##\dfrac{1}{ax+b}##

    This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?
  8. Delta2

    Integration by factors

    I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself. Any helpfull tips?
  9. tehsportsmaen

    I Two-Mass Oscillator: Plotting Amplitudes Over Frequencies in Hertz

    Idea: Given a system of two coupled oscillators in which 2 masses are connected to a spring in the middle. Each of the two masses is coupled to another spring on the left and right, which have fixed ends but are not connected to each other. So we have 3 springs, two masses and the springs also...
  10. G

    Help me prove integral answer over infinitesimal interval

    In the book, I see the following: ##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##. My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...
  11. MatinSAR

    Force field in spherical polar coordinates

    Picture of question: Part (a) : ##\nabla \times \vec F = 0## so a Potensial exists. I don't have problem with this part. Part (b) : what I've done : First experssion is 0 because ##\theta = \dfrac {\pi} {2}##. I don't know how to integrate over ##\theta ## when it is a constant.
  12. G

    I Integrating 1/x with units (logarithm)

    Hi. What exactly is happening mathematically when you integrate ##\frac{1}{x}## $$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$ if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...
  13. S

    Limit and Integration of ##f_n (x)##

    My attempt: (a) I don't think I completely understand the question. By "evaluate ##\lim_{n\to \infty f_n (x)}##", does the question ask in numerical value or in terms of ##x##? As ##x## approaches 1 or -1, the value of ##f_n (x)## approaches zero. As ##x## approaches zero, the value of ##f_n...
  14. Hamiltonian

    Finding a conditional probability from joint p.d.f

    using the equation mentioned under Relevant Equations I can get, $$\mathbb{P}(2X > Y |1 < 4Z < 3) = \frac{\mathbb{P}(2X>Y, 1<4z<3)}{\mathbb{P}(1<4z<3)}$$ I can find the denominator by finding the marginal probability distribution, ##f_{Z}(z)## and then integrating that with bounds 0 to 1. But I...
  15. Z

    Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##

    The characteristic equation has a zero discriminant and the sole root of ##-1##. The general solution to the associated homogeneous equation is thus $$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$ Now we only need to find one particular solution of the non-homogeneous equation. The righthand side of the...
  16. I

    A Integration of trigonometric functions

    Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
  17. Franyer

    I Integrate time dilation with derivative

    I need to integrate time dilation with derivative, how could I do that?
  18. P

    This integration appeared in the reconstruction of cross section

    I am reading the Horatiu Nastase's Introduction to quantum field theory (https://professores.ift.unesp.br/ricardo.matheus/files/courses/2014tqc1/QFT1notes.pdf ) ( Attached file ) or Peskin, Schroeder's quantum field theory book, p.105, (4.77). Through p.176 ~ p. 177 in the Nastase's Note, he...
  19. Valour549

    Two ways of integration giving different results

    I am trying to do the double integral. And I remembered there's this formula that says if the integrand can be split into products of F(x) and G(y) then we can do each one separately, then take the product of each result. Taken from Stewart's Calculus 9E. So I tried to do the integral two...
  20. E

    Work done for round trip when force is function of velocity

    I'd have no problem with this sort of problem if the force were a function of position. But here, I'm not sure where to go. Perhaps I'd start with an expression for the work done over an arbitrary distance if the force is given by ##g(v)##:$$W = \int_a^b g(v) \, dx$$ Not sure what to do next...
  21. azizlwl

    I'm getting the wrong answer for the Indefinite Integral of: (x^2+2x)/(x+1)^2

    ((x+1)^2 -1)/(x+1)^2 dx 1-1/(x+1)^2 dx Let u=x+1 1-1/u^2 du u+1/u +c (u^2+1)/u +c Not as answer given in the book.
  22. H

    Question re: Limits of Integration in Cylindrical Shell Equation

    I have managed to get the answer given by the textbook I'm referencing: 3π (∛4) (1 + 3∛3) However, this took multiple attempts, as I was initially trying to integrate within domain x = 0 - 2. This is the area for the bit that's above the x-axis (y=0 as specified). But the above answer is...
  23. chwala

    I Integration of ##e^{-x^2}## with respect to ##x##

    My first point of reference is: https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2 I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial...
  24. chwala

    Differentiate the given integral

    My take: $$\int_{x^2}^{2x} \sin t \, dt$$ using the fundamental theorem of calculus we shall have, $$\int_{x^2}^{2x} \sin t \, dt=-2x \sin x^2 +2 \sin 2x$$ I also wanted to check my answer, i did this by, $$\int [-2x \sin x^2 +2 \sin 2x] dx$$ for the integration of the first part i.e...
  25. TGV320

    I Integration formula of cosecant

    Hi I have a question about the integration formula of cosecant which leaves me puzzled. I usually find it written as " = ln |csc x - cot x| + C" in most manuals, but sometimes it is written as "= - ln |csc x + cot x| + C" or "= - ln (csc x + cot x) + C". Why is that? Can they all be...
  26. chwala

    Solve the given problem that involves integration

    For part (a), Using partial fractions (repeated factor), i have... ##7e^x -8 = A(e^x-2)+B## ##A=7## ##-2A+B=-8, ⇒B=6## $$\int {\frac{7e^x-8}{(e^x-2)^2}}dx=\int \left[{\frac{7}{e^x-2}}+{\frac{6}{(e^x-2)^2}}\right]dx$$ ##u=e^x-2## ##du=e^x dx## ##dx=\dfrac{du}{e^x}## ... also ##u=e^x-2##...
  27. 1

    Integration Substitution Techniques for quadratic expressions under square roots

    Hi, With respect to the techniques mentioned in point 2 and 3: Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques. Below excerpt shows problems 4 and 5 referenced in the above 2 points:
  28. S

    Solving this definite integral using integration by parts

    Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks
  29. C

    Impulse integration for a Tennis Racket hitting a Tennis Ball

    For this, Can someone please tell me why they integrate the impulse over from ##t_i## to ##t_f##? Why not from ##j_i## to ##j_f##? It seems strange integrating impulse with respect to time. Many thanks!
  30. A

    B Explain Integration to me, please

    Hi all, I understand what the integral does - it calculates the area under a curve and can easily see how it could be used to calculate an area of land. What I do not understand is really the physical meaning when it comes to the real world. Here are some examples: 1. A set of data...
  31. fresh_42

    Insights Can You Solve the Mathematical Mystery Behind the Art of Integration?

    This article cannot replace the 1220 pages of the almanac Gradshteyn-Ryzhik but it tries on 1% of the pages to summarize the main techniques. Continue reading...
  32. kirkr

    Solution to Schwarzschild Equation for Constant t,r

    In 1916, Karl Schwarzschild was the first person to present a solution to Einstein's field equations. I am using a form of his equation that is presented in Tensors, Relativity and Cosmology by Mirjana Dalarsson and Nils Dalarsson (Chapter 19, p.205). I am approaching what may be the simplest...
  33. fresh_42

    Insights An Overview of Complex Differentiation and Integration

    I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments that can be found in many excellent standard textbooks. Analysis is about differentiation. Hence, complex differentiation will be my starting point...
  34. E

    I Integration of Bessel function products (J_1(x)^2/xdx)

    Hello, While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral ## \int_0^\infty J_1(x)^2\frac{dx}{x}=1/2 ## I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
  35. O

    Symbolic integration of a Bessel function with a complex argument

    Hello all I am trying to solve the following integral with Mathematica and I'm having some issues with it. where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by Where delta is a coefficient. Due to the complex arguments I'm integrating the...
  36. E

    I Parameter Integration of Bubble Integral

    Referring to this link : https://qcdloop.fnal.gov/bubg.pdf Using Mathematica Integrate command to solve it does not give the result stated here but I am unclear as to how they got to the result in the 4th line. It is clear that the integrand (1st line) can diverge for certain values of the...
  37. chwala

    Applying integration to math problems

    Ok i know that, ##\int (x+2)^2 dx= \int [x^2+4x+4] dx= \dfrac{x^3}{3}+2x^2+4x+c## when i use substitution; i.e letting ##u=x+2## i end up with; ##\int u^2 du= \dfrac{u^3}{3}+c=\dfrac {(x+2)^3}{3}+c=\dfrac{x^3+6x^2+12x+8}{3} +c## clearly the two solutions are not the same... appreciate your...
  38. N

    I Calculate Length Contraction for Accelerated Motion to Proxima Centauri

    Let's assume a spaceship traveling from the Earth to the Proxima Centauri with constant acceleration g = 9.81 m/s2. The ship is accelerating the first half of the trajectory and decelerating the second half. I calculated the velocity profile from the Earth reference: The travel time on...
  39. S

    Question about approximate numerical integration methods

    This isn't a homework question per se but I can post more details like the data points & my work after. Suppose we are given a set of arbitrary points for which we cannot find an equation and we need to find the area under the curve without an analytical method - we can use either of the three...
  40. mncyapntsi

    Confused about polar integrals and setting up bounds

    So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds. Any help would be greatly appreciated! Thanks!!
  41. Eclair_de_XII

    I Relating integration of forms to Riemann integration

    Partition each closed interval ##[a_i,b_i]## in the Cartesian product, ##A##. Denote the partition for the i-th closed interval as ##\{x_i^1,\ldots,x_i^{k_i}\}##. The Cartesian product of the partitions forms a partition of ##A## (think: a lattice of points that coincide with the points of each...
  42. C

    No Limits of Integration for Electric Field Integral?

    For this problem, The solution is, However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q. However, how would you solve this problem with limits of integration? Many thanks!
  43. murshid_islam

    I Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##

    I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check? Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
  44. chwala

    Find ##f(x)## in the problem involving integration

    Q. 3(b). This is a textbook problem; unless i am missing something ...the textbook solution is wrong! solution; Mythoughts; ##f(x)=2\cos 3x-3\sin 3x## ...by using the product rule on ##\dfrac{d}{dx} (e^{2x} \cos 3x)##.
  45. Q

    Integration of structure function F2 to calculate quark momentum

    I study particle physics with “Particles and Nuclei” / Povh et al. and “Modern particle physics” / Mark Thomson and I am currently at “Deep-Inelastic scattering”. After introducing several scattering equations, such as Rosenbluth, that all include terms for electric AND magnetic scattering, i.e...
  46. U

    Complex Integration Along Given Path

    From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
  47. F

    Jacobian: how to change limits of integration?

    Hello, I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##. From there, I can reformulate with respect to ##z## and...
  48. T

    Integration of acceleration in polar coordinates

    I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image. My attempts are the following, I proceed using 3 "independent" methods just as you...
  49. A

    I Integration of √(1-x^2) with x=sinθ: Check Correctness and Simplification

    Does my below integration is correct? ##\int \sqrt{1 - x^2} \ dx## Let ##x = \sin \theta##, then ##dx = \cos \theta \ d \theta##, ##\cos \theta = \sqrt{1 - x^2}##, ##\theta = \sin^{-1} (x)## ##\int \sqrt{1 - x^2} \ dx## ##= \int \left( \sqrt{1 - \sin^2 \theta} \right) \ \left( \cos \theta \...
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