Integration Definition and 1000 Threads

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. A

    Calculus Textbook for Integration using Hyperbolic substitution

    Can someone please tell me the book that contain integration using hyperbolic substitution for beginner? I know that hyperbolic functions is taught in Calculus book but most of them is only some identities and inverses of hyperbolic functions.
  2. ergospherical

    I PF Integral Bee: Share Interesting/Quirky Integrals!

    Thought it could be fun to have a sort of "PF Integral Bee"... if you know some interesting/quirky/etc. integrals then post them here! 🤓 To get the ball rolling... 1. ##\displaystyle{\int_0^1} \dfrac{\ln{(x+1)}}{x^2+1} dx##
  3. A

    I Integration Using Hyperbolic Substitution

    Can someone please show me an example of integration using hyperbolic substitution? Thank you
  4. Istiak

    How integral and gradient cancels?

    I know that gradient is multi-variable derivatives. But, here line integration (one dimensional integral) had canceled gradient. How?
  5. Istiak

    How to find the constant in this indefinite integration?

    $$x(t)=\int \dot{x}(t)\mathrm dt=vt+c$$ That's what I did. But, book says $$x(t)=\int \dot{x}(t)\mathrm dt=x_0+v_0 t+ \frac{F_0}{2m}t^2$$ Seems like, $$x_0 + \dfrac{a_0}{2}t^2$$ is constant. How to find constant is equal to what?
  6. Safinaz

    Integration of an exponential function

    My trial : I think ## \int ~ dy ~ e^{-2 \alpha(y)} ## dose not simply equal: ## - \frac{1}{2}e^{-2 \alpha(y)} ## cause ##\alpha## is a function in ##y ##. So any help about the right answer is appreciated!
  7. bob012345

    I Exact Integration of Newton's Gravitational Law?

    I realized I never actually derived the kinematic equations of motion for the exact Newtonian gravitational force. For an object falling near the surface of the earth, how do we handle integrating the equation of motion to derive the kinematics equations without using the approximation of...
  8. M

    Monte-Carlo integration does not work properly after implementing pdfs

    Does anyone have experience with such strange behavior in Monte-Carlo methods? I think it is a conceptional problem and I am just missing a key point in how to set up the integration instead of a error in my code itself. I use data files from LHAPDF and also checked that my variables give the...
  9. DaalChawal

    MHB Integration Doubt: Answers & Solutions

  10. Eclair_de_XII

    Converting integration of rectangular integral to spherical.

    I'm going to type out my LaTeX solution later on. But in the meantime, can anyone check my work? I know it's sloppy, disorganized, and skips far more steps than I care to count, but I'd very much appreciate it. I'm not getting the answer as given in the book. I think I failed this time because I...
  11. R

    Calculating Potential Energy from Force for Non-Linear Systems

    If I have a force that behaves according to the formula ##F(x)=\alpha x-\beta x^3##, how can I get the potential energy from it? I know that: $$-\frac{\mathrm{d}V(x)}{\mathrm{d}x}=F(x),$$ but what about the limits of the integration?
  12. J

    MHB Integration Help: Struggling with Distance Qn

    I’ve always struggled with integration and I don’t know how to do this question, I’m not sure what I’m being asked to calculate. I tried to calculate this as a definite integral but there is no boundary conditions for the distance the object has traveled which is confusing any help would be...
  13. A

    Integration by filaments or integration by strate?

    Greetings While solving the following exercice, ( the method used is the integration by filaments and I have no problem doing it this way) here is the solution My question is the following: I want to do the integration by strate and here is my proposition is that even correct? I would like...
  14. A

    Changing the order of integration

    Greetings! As mentionned my aim is to change the order of integral, and I totally agree with the solution I just have one question: as you can see they have put 0<=y<=1 and 0<=x<=y^2 but would it be wrong if I put 0<=y<=1 and y^2<=x<=1? Thank you!
  15. C

    Using params from gsl function in integration routine

    I'm trying to pass through some parameters of a function to the gsl integration routine but my code is currently not returning correct values. I attach a version of my code using dummy example functions and names. struct myStruct_t { double a; }; double func(double z, void* params)...
  16. JD_PM

    Rewriting a given action via integration by parts

    I simply plugged \phi = \phi_0 (\eta) + \delta \phi (\eta, \vec x) into the given action to get \begin{align} S &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi^2 -(\nabla \phi)^2\right)-a^4V(\phi) \right] \nonumber \\ &= \int d^4 x \left[ \frac{a^2}{2}\left(\dot \phi_0^2 + (\delta...
  17. greg_rack

    Volume of a solid of revolution around the y-axis (def. integration)

    First, I calculated the inverse of ##y=e^x## since we're talking about y-axis rotations, which is of course ##x=lny##. Then, helping myself out with a drawing, I concluded that the total volume of the solid must've been: $$V=\pi\int_{0}^{1}1^2 \ dy \ +(\pi\int_{1}^{e}1^2 \ dy \ - \pi...
  18. E

    Calculus Practical reference for integration on manifolds

    I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity. Tong's notes and some others are good on the abstract/theoretical side but it'd really be better at this stage to get some practice with concrete examples in order to...
  19. E

    I Integration trouble (integral over a 2-sphere)

    There's an integral over a 2-sphere ##S## with unit normal ##N^a## within a hypersurface orthogonal to a Killing field ##\xi^a##$$F = \int_S N^b (\xi^a / V) \nabla_a \xi_b dA = \frac{1}{2} \int_S N^{ab} \nabla_a \xi_b dA, \quad N^{ab} := 2V^{-1} \xi^{[a} N^{b]}$$which follows because the Killing...
  20. tanaygupta2000

    How Can Cylindrical Coordinates Simplify Complex Number Integration?

    I began this solution by assuming a = x+iy since a is a complex number. So I wrote expressions of <a| and |a> in which |n><n| = I. I got the following integral: Σ 1/πn! ∫∫ dx dy exp[-(x^2 + y^2)] (x^2 + y^2)^n I I tried solving it using Integration by Parts but got stuck in the (x^2 + y^2)^n...
  21. R

    Integration of abs(k)e^(ikx)dk

    Split the integral $$\frac{Aa}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{ikx}dk - \frac{A}{\sqrt{2\pi}}\int^{\infty}_{-\infty}|k|e^{ikx}dk$$ Apply the boundary conditions, this is where my biggest source of uncertainty comes from I doubled the integral and integrated from 0 to a instead of from -a...
  22. S

    Calculus Calculus textbooks with good sections on integration

    Hi I'm having troubles with integration specially by substitution, I'm going to read a calculus textbook and i need recommendations of books with a good treatment on the different techniques of integration. I'd like a book with good exercises for self study and a exposure to integration of...
  23. S

    Integration of this trigonometry function

    Is it possible to do the integration? That is the full question I don't know where to start, try to use ##u=\cos x## and also ##\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)## but failed. Thanks
  24. A

    Problem with setting the region of integration

    Good day ! I have a problem with the solution of the floowing integrals Indeed i don't understand why they choose such borders for integral b/a<c y<c doesn't mean that y<b/a ! many thanks in advance!
  25. Mayhem

    B Why don't we account for the constant in integration by parts?

    As we all know, integration by parts can be defined as follows: $$\int u dv = uv - \int v du$$ And the usual strategy for solving problems of these types is to intelligently define ##u## and ##dv## such that the RHS integral can easily be evaluated. However, something that is never addressed is...
  26. JD_PM

    Integration and hyperbolic function problem

    This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...
  27. S

    Understanding Griffith's Velocity Argument for Charge Integration

    In Griffith’s section 10.3.1, when proving why there is an extra factor in integrating over the charge density when it depends on the retarded time, he makes the argument that there can only ever be one point along the trajectory of the particle that “communicates” with the field point. Because...
  28. S

    B Integration of tan^2 x from - to + infinity

    ∫tan^2 x ( -infinity to +infinity)
  29. D

    I Help With a Proof using Contour Integration

    I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof. He is evaluating an integral along a contour which makes up the imaginary axis. He has the following: $$ -i\int_{-iR}^{iR}...
  30. S

    B Integration from "Area Under Curve" Perspective: Explained

    I can calculate the value of the integration, it will be ##\frac{\sqrt{3}}{2}## But if I draw the function and consider the area bounded by the curve and x-axis from x = 0.5 to x = 1, it seems that the area will be infinite because x = 1 is vertical asymptote. Why can't I consider from "area...
  31. M

    Integrating Partial Fractions with a Twist: (x + 3) vs (x - 3)

  32. chwala

    Integration of a trig function

    This is my first attempt ...
  33. chwala

    Integration of a trigonometic function

    my thinking was to have everything changed to a function that has cosine only... ##\int_0^{0.5π} \frac {1-cos^2x}{sin x + cos x}dx## ##\int_0^{0.5π} \frac {(1-cos x)(1+cos x)}{(1-cos^2x)^{0.5} + cos x}dx## ... first of all is this integration possible? if so then let me know if i am on the...
  34. Tony Hau

    I How to interpret integration by parts

    So I am confused about a proof in which the formula for expected value of velocity, ##\frac{d\langle x \rangle}{dt} ##, is derived. Firstly, because the expected value of the position of wave function is $$\langle x \rangle =\int_{-\infty}^{+\infty} x|\Psi(x,t)|^2 dx$$Therefore...
  35. R

    I Is it possible to solve such a differential equation?

    Hello, I would like to is it possible to solve such a differential equation (I would like to know the z(x) function): \displaystyle{ \frac{z}{z+dz}= \frac{(x+dx)d(x+dx)}{xdx}} I separated variables z,x to integrate it some way. Then I would get this z(x) function. My idea is to find such...
  36. WannaLearnPhysics

    Using Ampere's Law for these two different integration paths

    Homework Statement:: The magnetic field at every point on the path of integration Relevant Equations:: The scenarios/situations are shown in the attached photo. "Any conductors present that are not enclosed by a particular path may still contribute to the value of B field at every point, but...
  37. L

    Calculus and Kinematic equations--- seeing the logic

    Details of Question: ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into: s − s0 = v0t + ½at2 My main question is about the integration of...
  38. J

    I Integration of differential forms

    I am confused as to how exactly we integrate differential forms. I know how to integrate them in the sense that I can perform the computations and I can prove statements, but I don't understand how it makes sense. Let's integrate a 1-form over a curve for example: Let ##M## be a smooth...
  39. N

    Integration by parts on ##S^3## in Coleman's textbook

    I'm reading Coleman's "Aspects of symmetry" chap 7. On the topic of the SU(2) winding number on ##S^3##on page 288, three parameters on ##S^3## are defined ##\theta_1,\theta_2,\theta_3##. Afterwards, it defines the winding number and to show it's invariant under continuous deformation of gauge...
  40. P

    A What is the meaning of ##d\Omega## in solid angle integration?

    Anyone have any idea how to perform the following two integrals? ##\int d\Omega n_{i}n_{j}## and ##\int d\Omega n_{i}n_{j}n_{k}n_{l}## where the n is a unit vector.
  41. B

    Integrating with a Denominator of (1+x^2)

    I think in the case of "n da" you can see the denominator (1+x^2) as a constant, so ∫ ( sin(a) + M^2 ) / ( 1 + x^2 ) da = ( 1 / ( 1 + x^2 ) ) * ∫ (sin(a) + M^2 ) da = ( 1 / ( 1 + x^2 ) ) * ( -cos(a) + (M^2)a ) = ( - cos(a) + (M^2)a ) / ( 1 + x^2 ) --- Is this the way to go? This is my...
  42. T

    I Bose-Einstein numerical integration

    Want to integrate the total energy density over all photon energies between two temperature values from 500K to 5800K, but not sure how to proceed. Here is some examples to help:
  43. jaychay

    MHB Can Integration by Parts Solve This Tricky Question?

    Can you help me with this question ? I am really struck with this question. Thank you in advance.
  44. jaychay

    MHB Can You Help Me Integrate $\sin(u) + u^6$ Correctly?

    Can you please help me ? I have tried to do it many times but still got the wrong answer. Thank you in advance.
  45. H

    Integration ## f(\theta, \phi) = \frac{sin \theta}{4\pi}##

    Hi, I have this formula ## f(\theta, \phi) = \frac{sin \theta}{4\pi}## I have this statement that say if I integrate this formula above on a sphere then p = 1. what does integrate on a sphere means? I know ##\int_0^{2\pi} ## is used for the circle.
  46. BWV

    A Integration with respect to a Lévy basis / Ambit fields

    Familiar with basics of stochastic calculus and integration over a Brownian motion. Trying to get a sense of Ambit Fields https://en.wikipedia.org/wiki/Ambit_field which mention an integration over a Lévy basis: Curious if anyone familiar with this? A Brownian motion is a Levy process...
  47. greg_rack

    B How to learn differentiation and integration in 14 days?

    The detailed list of the concepts I should master I'm attending the last year of high school and I'm currently studying limits. For university test reasons I'll need to study on my own topics such as differentiation and integration... and I have just 14 days to do so! Firstly, do you think it's...
  48. D

    Reversing the order of integration in a double integral

    Performing the x-integration first the limit are x=y2 and x= -y2 and then the y limits are 0 to 1. This gives the final answer 2/5 But i am getting confused when trying to reverse the order of integration. My attempt is that i have to divide the region in 2 equal halfs and then double my answer...
  49. H

    Integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}##

    Hi, I'm trying to integrate ## \int \frac{zdz \cdot \hat z}{(\sqrt{p^2 + z¸^2)^3}}## and ## \int \frac{pdz \cdot \hat p}{(\sqrt{p^2 + z¸^2)^3}}## I get ## \frac{\hat z}{(\sqrt{p^2 + z¸^2)}}## and ## \frac{2z \cdot \hat p}{(\sqrt{p^2 + z¸^2)}}## But the correct answer should be ## \frac{z \hat...
  50. H

    Integration ##\ddot\phi = -\omega^2\phi##

    Hi, I'm wondering how can I get ## \phi(t) = A sin(\omega t) + B cos(\omega t)## I know I have to integrate 2 times ##\ddot\phi = -\omega^2\phi##. However, I don't have any more explanation in my book. I know A and B are the constants of integration.
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