Integration Definition and 1000 Threads

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...

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...

((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.

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...

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...

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...

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...

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##...

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:

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

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!

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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!!

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...

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!

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}##...

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)##.

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...

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 +...

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...

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...

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 \...

In a textbook I own a formula is given for the integration of natural coordinates over an element. In this case it is a 1 dimensional element (i.e. a line segment) with coordinates ##x_i## and ##x_j##. The coordinate ##x## over the element is written as: $$ x = L_1(x) x_i + L_2(x) x_j $$ with...

The note is entitled: Evaluation of a Class of n-fold Integrals by Means of Hadamard Fractional Integration. 4 pgs pdf format. I assure you that you need not know anything about fractional calculus at all to understand this note that Howard Cohl helped me with. We only use a single...

[FONT=times new roman]Problem Statement : [FONT=times new roman]To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP.[FONT=times new roman] Attempt 1 (without calculus): I mark some relevant lengths inside the circle...

##\langle T(f), g \rangle = \int_{0}^{1} \int_{0}^{x} f(t) dt ~ g(t) dt## As ##\int_{0}^{x} f(t) dt## will be a function in ##x##, therefore a constant w.r.t. ##dt##, we have ##\langle T(f), g \rangle = \int_{0}^{x} f(t) dt ~ \int_{0}^{1} g(t) dt## ##\langle f, T(g)\rangle = \int_{0}^{1} f(t)...

Hi, you all, I open this thread to ask for any recommendation concerning integration as well as ODE/PDE solving techniques for physics. I love mathematics, and I usually read material on pure mathematics (most notably abstract algebra and a bit of topology) but here I'm more interested in the...

If we solve the L.H.S. of this equation, we get ## \frac{(b-a)^3}{6}## and if we solve R.H.S. of this equation, we get ##-\frac{2b^3-3ba^2 +a^3}{6}## So, how can we say, this equation is valid? By the way, how can we use the hint given by the author here?

Initially '0' is the upper limit and ##a = \frac{Ze^2}{E}## is the lower limit. With change of variable ##x = \frac{Er}{Ze^2}##, for ##r=0##, ##x=0##, and for ##r=\frac{Ze^2}{E}##, ##x=1##, so 1 should be the lower limit. However, he takes 1 as the upper limit, and without a minus sign. Why is...

How can i move from this expression: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+i(k-k_{f}))^3} \frac{1}{(1+i(k-k_{i}))^3}$$ to this one: $$\frac{4}{\pi^{4}} \int dk \frac{1}{k^2} \frac{1}{(1+|k-k_{i}|^2)^2} \frac{1}{(1+|k-k_{f}|^2)^2}$$ using Feynman parametrization (Integration by...

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Will it be [{(r-a)/r}*H(r-a)]

Hi PF! I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this NIntegrate[-(1/((-1.` + x)^2 (1.` + x)^2 (1.` + y)^2)) 3.9787262092516675`*^14 (3.9999999999999907` + x (-14.99999999999903` + x (20.00000000000097` -...

https://www.wuft.org/news/2022/02/18/buchholz-high-school-student-discovers-and-publishes-new-calculus-technique/

I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ... I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of...

I'm not sure where to start, when I tired using integration of the initial equation to get pos(t)=-.65t^2 i + .13t^2 j + 14ti +13tj but after separating each component, i and j, and setting j equal to zero I got 0 or -100 seconds which doesn't seem like a reasonable answer.

Integration of v= integration of##(alpha \sqrt x)dx##. But I am getting wrong answer.

If we have system of 3 ordinary differential equation in mechanics and we have two initial condition ##\vec{r}(t=0)=0## and ##\vec{v}(t=0)=\vec{v}_0 \vec{i}##. If we somehow get \frac{d^2v_x}{dt^2}=-\omega^2v_x then v_x(t)=A\sin(\omega t)+B\cos(\omega t) Two integration constants and one initial...

I tried using substitution ##u=\sqrt{16x-x^8}##, didn't work Tried factorize ##x## from the denominator and then used ##u=\sqrt{16-x^7}##, didn't work Tried using ##u=x^4## also didn't work How to approach this question? Thanks

Hello, I would like help to clarify what det( {\delta \over \delta J}) W(J) (equation 15.79) actually means, and why it returns a number (and not a matrix). This comes from the following problem statement (Kaku, Quantum Field Theory, a Modern Introduction) Naively, one would define det...

does anyone know how to solve this/can lead me on a direction to where I will get an answer that actually makes sense lol? I keep getting a negative answer/0. For context, I'm tryna find the surface area of a pringle and all the sources I've visited always estimate the projected 2D region as a...