Integral Definition and 1000 Threads

Homework Statement if ## f(x) ={\int_{\frac{\pi^2}{16}}^{x^2}} \frac {\cos x \cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz## then find ## f'(\pi)## 2. The given solution Differentiating both sides w.r.t x ##f'(x) = {-\sin x {\int_{\frac{\pi^2}{16}}^{x^2}} \frac{\cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz }+{...

Homework Statement solve ##\int_0^1 x^6 \arcsin{x} dx##

Homework Statement If ## I_n = \int_0^\frac {\pi}{4} \sec^n x dx## then find ## I_{10} - \frac {8}{9} I_8## 2. The attempt at a solution this should be solvable by reduction formulae but since it'd be longer I wanted to know if there was a way to do it using mostly properties of indefinite...

Homework Statement evaluate the following definite integral with limits 0 to 1## ∫log(sin(πx/2)) dx ## 2. The attempt at a solution I used ##∫f(x) = ∫f(a+b-x)## to get ## I=∫log(cos(πx/2))## with the same limits. Adding them and using ##log(m)+log(n)=log(mn)## and ##2sinxcosx=sin2x## I got...

I am looking at a solution to an integral using differentiation under the integral sign. So let ##\displaystyle f(t) = \frac{\log (tx+1)}{x^2+1}##. Then, through calculation, ##\displaystyle f'(t) = \frac{\pi t + 2 \log (2) - 4 \log (t+1)}{4(1+t^2)}##. The solution immediately goes to say that...

Hello, How can I evaluate the following nested integral in MATLAB for a general value of ##K## {\int\limits_{u_1=0}^{\gamma}\int\limits_{u_2=0}^{\gamma-U_1}\cdots \int\limits_{u_{K}=0}^{\gamma-\sum_{k=1}^{K-1}U_k}}f_{U_1}(u_1)f_{U_2}(u_2)\cdots f_{U_{K}}(u_{K})\,du_1du_2\cdots du_{K} where...

In this super short video of the derivation of the relativistic kinetic energy, , I'm just stuck on one thing. Around 1:00 minute in, the constants of integration change from 0 to pv when the integration changes from dx to dv. Where does the pv come from? Thanks!

Homework Statement I have the following expression $$I = \int_{-\infty}^{0} f_p(p) \ \big[ pf_x(a - \frac{p}{m}t) \big] dp + \int_{0}^{\infty} f_p(p) \ \big[ pf_x(a - \frac{p}{m}t) \big] dp$$ where ##f_p## and ##f_x## are normalised distributions. In particular, ##f_x## is symmetric about...

Homework Statement Show that u(x, y) = y/π ∫-∞∞ f(t) dt / ((x - t)2+y2) satisfies uxx + uyy = 0. Homework Equations Leibniz' Rule The Attempt at a Solution I'm not even sure Leibniz' Rule can be applied here since there seems to be a discontinuity in the integrand when x=t and y=0. When I...

Homework Statement Prove the integral of x*arcsine(x) from 1/2 to 1 is bounded between pi/16 and 3*pi/16 Homework EquationsThe Attempt at a Solution Not sure what to bound with. Do we use Squeeze Theorem?

This is actually a WolframAlpha question, but I suppose someone conversant in mathematica could give me an answer. How in Mathematica could I compute ##\displaystyle \int_0^1 \left( \prod_{r=1}^3 (x+r)\right) \left(1+x \sum_{r=1}^3 \frac{1}{x+r} \right) ~ dx##. I tried int (Product[x+r, {r...

Homework Statement Find the centre of mass of a uniform hemispherical shell of inner radius a and outer radius b. Homework Equations ##r_{CoM} = \sum \frac{m\vec{r}}{m}## The Attempt at a Solution Using ##x(r,\theta,\phi)## for coordinates...

Hello everyone! I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...

Hello, I've recently discovered the sine integral and have been playing around with it a bit on some graphing software. I looked at the graph of ##Si(x^2) - \frac π 2## and saw that both the amplitude and period was decreasing as x increased. Curiosity got the best of me so I decided to...

Say we have a difficult integral of the form ##\displaystyle \int_a^{b}f(x) ~dx##. Let ##t = \frac{x-a}{b-x}##. Then ##\displaystyle \int_0^{\infty}f \left( \frac{bt+a}{t+1} \right)\frac{1-a}{(t+1)^2} ~dt##. My idea is that making this change of variables transforms the integral into a form...

Homework Statement ##\displaystyle \int_0^{\infty} \frac{\log (x+1)}{x(x+1)} dx## Homework EquationsThe Attempt at a Solution I tried to convert the log to a series, but that got be nowhere, since the resulting integral was divergent. Any hints on how to approach this?

I have the integral ##\int_{-\infty}^{\infty} x^2 e^{-x^2} ~dx##. Is there any simple way to integrate this, given that that I already know that the value of the Gaussian integral is ##\sqrt{\pi}##?

Hey, I tried to construct the derivation of the integral C with respect to Y: $$ \frac{\partial C}{\partial Y} = ? $$ $$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$ with $$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...

Convolution has the form (f\star g)(t) = \int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau However, I for my own purposes I have invented a similar but different type of "convolution" which has the form (f\star g)(t) = \int_0^{\infty}f(\tau)g(t/\tau)d\tau So instead of shifting the function g(t)...

Homework Statement ##\displaystyle \int_0^1 \frac{\arctan x}{x}dx## Homework EquationsThe Attempt at a Solution I converted the integral to the following; ##\displaystyle \int_0^1 \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{2n+1}dx##. In this case am I allowed to swap the summation and integral signs?

I have a double integral where the region of integration is ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. I am doing the change of variables ##u=zt## and ##v = z(1-t)##. I am a bit rusty on calc III material, so how would I find the new region of integration, in terms of the...

Homework Statement The path integral from (0,0,0) to (1,1,1) of $$<x^2,2yz,y^2>$$. I am a little confused about the setup.Homework Equations $$\int_{a}^{b} v.dl$$The Attempt at a Solution Here is how I set it up. $$\int_{0}^{1}x^2 dx + \int_{0}^{1}2yz dy + \int_{0}^{1}y^2 dz$$ Since the...

Let $\lambda$ be a positive number and $n$ a natural number. I want to show that $$\int_{-\infty}^{+\infty} x^{2n} e^{-2 \lambda x^2} dx<+\infty.$$ There is the following hint: $e^{\lambda x^2} \geq \frac{1}{n!}(\lambda x^2)^n$, thus $x^{2n} e^{-\lambda x^2}\leq \frac{n!}{\lambda^n}$.I have...

Homework Statement ##\displaystyle \int_0^1 \frac{1}{\lfloor 1- \log_2 (1-x)\rfloor}## Homework EquationsThe Attempt at a Solution How in general do I approach integrals with floor functions? I'm thinking maybe I can figure out what the function in the integrand looks like on paper and...

Homework Statement ##\displaystyle \int \sin x \sqrt{1+ \tan ^2 x} dx## Homework EquationsThe Attempt at a Solution So clearly we have that ##\displaystyle \int \sin x \sqrt{\sec ^2 x} dx##, but I am not sure how to proceed. Isn't it true that ##\sqrt{\sec ^2 x} = | \sec x|##? How would I...

I have the integral ##\displaystyle \int_{- \infty}^{\infty} \frac{\cos x}{x^2+1} dx##. We are going to use differentiation under the integral sign, so we let ##\displaystyle I(t) = \int_{- \infty}^{\infty} \frac{\cos tx}{x^2+1} dx##, and then, after manipulation, ##\displaystyle I'(t) = \int_{-...

Homework Statement ##\displaystyle \int \frac{1}{x^2+a} dx## Homework EquationsThe Attempt at a Solution I know that I can convert this to the form ##\displaystyle \int \frac{1}{x^2+(\pm \sqrt{a})^2} dx## = ##\displaystyle \frac{1}{\pm \sqrt{a}} \arctan (\frac{x}{\pm \sqrt{a}}) + C##, but I...

Hey! :o Let $u(x,t), A(x)$ be functions, for which holds the following: We have the pde $u_t+a(u)u_x=0$. Let $A'(u)=a(u)$ then the pde can be written as $u_t+A(u)_x=0$. We have the following integrals $$\int_{a-\epsilon}^au\cdot \left (\frac{x-a}{\epsilon}+1\right )\...

In the table of integral, there is this formula \int_0^{\infty}\frac{sin(ax)}{x}\,dx=\frac{\pi}{2}\text{sign} a is sign a here is literally the sign of a, or it means something else?

I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?

Hello all, I am reading through the Jackson text as a hobby and have reached a question regarding the Hamiltonian transformation properties. I will paste the relevant section from the text below: I don't understand what he's getting at in the sentence I highlighted. To attempt to see what...

Homework Statement I am studying Gerry's <Introductory Quantum Optics>, in which there is an integral (Eq. 4.37) $$\intop_{-infinity}^{+infinity}\frac{[sin(\triangle t/2)]^{2}}{\triangle^{2}}d\triangle=\frac{\pi}{2}t.$$ I don't know how to get the result of the right side. Homework Equations I...

Hey! :o Let $I=[a,b]$, $J=[c,d]$ compact intervals in $\mathbb{R}$, $g,h:I\rightarrow J$ differentiable, $fI\times J\rightarrow \mathbb{R}$ continuous and partial differentiable as for the first variable with continuous partial derivative. Let $F:I\rightarrow \mathbb{R}$. I want to calculate...

Homework Statement Refer the image. Homework Equations Integral (e^x)dx from 0 to 1=e-1.The Attempt at a Solution Refer the other image. The graph of e^(x^2) increases more slowly than e^x till x=1. So 'a' is clearly greater than 1. Is this right?

Sorry if this is in the wrong place. Sometimes I do really stupid things on integrals (use a method that gets me nowhere, make a mistake while factoring quickly, etc.) I have always been reluctant on using tables because I always felt stupid using them. I feel like I have to reinvent every...

Hello everyone Can someone help me out solving this integral: \begin{equation} S_T(\omega)=\frac{2k_BT^2g}{4\pi^2c^2}\int_0^{\infty}\frac{sin^2(kl)}{k^2l^2}\frac{k^2}{D^2k^4+\omega^2}dk \end{equation} Where $$D=g/c$$ According to this paper https://doi.org/10.1103/PhysRevB.13.556. The...

Give your answer in exact form. Format ±X/Y ± is if the value is positive or negative (Duh) X is the numerator (e.g. 7) Y is the denominator (e.g. 11) the question:

Homework Statement Show that \int_{A} 1 = \int_{T(A)} 1 given A is an arbitrary region in R^n (not necessarily a rectangle) and T is a translation in R^n. Homework Equations Normally we find Riemann integrals by creating a rectangle R that includes A and set the function to be zero when x...

How may we go about to show that, $$\int_{0}^{1}t\cos(2t\pi)\tan(t\pi)\ln[\sin(t\pi)]\mathrm dt=\color{green}{1\over \pi}\cdot\color{blue}{{\ln 2\over 2}(1-\ln 2)}$$

How to prove this integral, $$\int_{0}^{2\pi}\sin\left({x\over 2}\right)\ln^2\left[\sin\left({x\over 4}\right)\sin\left({x\over 8}\right)\right]={27\over 2}+\ln^2(2)+\ln(2)$$

How to prove this integral, $$\int_{0}^{2\pi}\mathrm dt{\sin t\over \sin t+ i\sqrt{n+\cos^2 t}}={\pi\over 1+n}$$ $n \ne -1$ $i=\sqrt{-1}$

Hi PF! I am integrating the following sin\[Theta][x_, \[Alpha]_] := Sqrt[(2 - 2 x^2)/( 3 - 4 x Cos[\[Alpha]] + Cos[2 \[Alpha]])] cos\[Theta][x_, \[Alpha]_] := Sqrt[(Cot[\[Alpha]] - x Csc[\[Alpha]])/( 1 + 2 x Cot[\[Alpha]] Csc[\[Alpha]] - 2 Csc[\[Alpha]]^2)] \[Rho][x_, \[Alpha]_] :=...

I am looking at the integral $$\int_0^\infty dx \: e^{-iax} - e^{iax}$$ I know that this does not converge for many reasons, but most obviously because I can rewrite it as $$2i \int_0^\infty dx \: sin(ax) = -2i a [\cos(ax)]_0^\infty$$ which does not converge to anything. However the book...

Homework Statement I am supposed to compute the Fourier transform of f(x) = integral (e-a|x|) Homework Equations Fourier transformation: F(p) = 1/(2π) n/2 integral(f(x) e-ipx dx) from -infinity to +infinity The Attempt at a Solution My problem is, that I do not know how to handle that there...

Homework Statement Use cylindrical coordinates to evaluate triple integral E (sqrt(x^2+y^2)dv where E is the solid that lies within the cylinder x^2+y^2 = 9, above the plane z=0, and below the plane z=5-y Homework EquationsThe Attempt at a Solution So i just need to know how to get the bounds...

Given: A so-called complicate integral has a such a simple closed form, quite amazed me, but how to prove it, is an other story. $$\int_{0}^{1}\mathrm dt{t-3t^3+t^5\over 1+t^4+t^8}\cdot \ln(-\ln t) dt=\color{red}{{\pi\over 3\sqrt{3}}}\cdot \color{blue}{\ln 2\over 2}$$ Does anyone know to how...

There is nothing wrong with the well known $$e^{i\theta}=\cos\theta+i\sin\theta$$ for real ## \theta## but what about $$\int_{-\infty}^\infty~e^{i\theta(p)}\mathrm{d}p=\int_{-\infty}^\infty~\cos\theta(p)\mathrm{d}p+i\int_{-\infty}^\infty~\sin\theta(p)\mathrm{d}p$$ I have been trying to use...

Homework Statement We have the equation for gravity due to the acceleration a = -GM/r2, calculate velocity and position dependent on time and show that v/x = √2GM/r03⋅(r/r0-1) Homework Equations x(t = 0) = x0 and v(t = 0) = 0 The Attempt at a Solution v = -GM∫1/r2 dt v = dr/dt v2 = -GM∫1/r2...

Proposed: How can we prove $(1)?$ $$\int_{0}^{\infty}\mathrm dx{\sin^2\left({a\over x}\right)\over (4a^2+x^2)^2}={\pi\over (4a)^3}\tag1$$

$$\int_{0}^{\pi\over 2}{\ln(\sin^2 x)\over \sin(2x)}\cdot \sqrt[5]{\tan(x)}\mathrm dx=-5\phi \pi$$ $\phi$ is the golden ratio Any help, please. Thank you!