Integral Definition and 1000 Threads

Homework Statement Compute the Integral: ##\int_{-\infty}^\infty \space \frac{e^{-2ix}}{x^2+4}dx## Homework Equations ##\int_C \space f(z) = 2\pi i \sum \space res \space f(z)## The Attempt at a Solution At first I tried doing this using a bounded integral but couldn't seem to get the right...

Homework Statement Hi guys, I am going to calculate the following integral: $$\int_0^{f_c+f_m} |Y(f)|^2\, df$$ where:$$Y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega +...

Homework Statement use the residue theorem to find the value of the integral, integral of z^3e^{\frac{-1}{z^2}} over the contour |z|=5 The Attempt at a Solution When I first look at this I see we have a pole at z=0 , because we can't divide by zero in the exponential term. and a pole of...

Homework Statement Hi everybody! I am asked to calculate how much of the total radiated power of a light bulb at temperature ##T=2300##K is contained within ##400##nm and ##750##nm. I am also given the average emissivity of tungsten ##\epsilon_\text{ave}=0.288## and the emissivity within the...

Homework Statement Homework Equations Summs: $$1+2+3+...+n=\frac{n(n+1)}{2}$$ $$1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$ The Attempt at a Solution $$\Delta x=\frac{b}{n}$$ $$S_n=f\left( \frac{\Delta x}{2} \right)\Delta x+f\left( \Delta x+\frac{\Delta x}{2} \right)\Delta x+...+f\left(...

Homework Statement Hi all, I hope you all can help me so I'm studying for my signals course and I encounter this example in the book, and the answer is there but the solution isn't... The convolution integral exists for 3 intervals and I could evaluate the first two just fine... however I can't...

Hello, I'm having some trouble with my octave coding and would appreciate any input on where the issue lies. The coding is as follows: age = [0:1:100]; %this is the age matrix, represented by a time = [0:1:100]; %this is the time matrix...

Hi everyone, my friend challenged me to solve this definite integral...integral from -2pi to 2pi ((sin(2sinx)+cos(2cosx))dx, i proved by using definite integral properties that this integral equals to integral from -2pi to 2pi cos(2cosx)dx, can you give me any ideas how to solve this?? I know...

Homework Statement WE have a thermally insulated metallic bar (from enviroment/surroundings) . It has a temperature of 0 ºC. At t=0 two thermal sources are applied at either end, the first being -10 ºC and the second being 10 ºC. Find the equation for the temperature along the bar T(x,t), in...

Homework Statement For the vector field F(r) = Ar3e-ar2rˆ+Br-3θ^ calculate the volume integral of the divergence over a sphere of radius R, centered at the origin. Homework Equations Volume of sphere V= ∫∫∫dV = ∫∫∫r2sinθdrdθdφ Force F(r) = Ar3e-ar2rˆ+Br-3θ^ where ^ denote basis (unit vectors)...

Homework Statement I am trying to evaluate the following integral: ##\displaystyle \int^{\infty}_0 (1 - e^{-x}) x^{-\frac{3}{2}} \, dx## Homework EquationsThe Attempt at a Solution When I split the above integral, I get the following ##\int^{\infty}_0 x^{-\frac{3}{2}} \, dx - \Gamma...

The integral formulation of Maxwell equations, for example the one called "Gauss' theorem", see the first equation here: https://en.wikipedia.org/wiki/Maxwell's_equations is still valid in dynamics, with rapidly varying sources? We know that a rapidly varying charge here gives a flux of an...

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...

Homework Statement Show that ##\mathbb{Z} [\sqrt{d}] = \{a+b \sqrt{d} \ | \ a,b,d \in \mathbb{Z} \}## is an integral domain Homework EquationsThe Attempt at a Solution Do I have to go through all of the axioms to do this? For example, do I have to show that it is an abelian group under...

$\tiny{242.tr.05}$ Use the integral test to determine if a series converges. $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sqrt{e^{2n}-1}}$ so... $\displaystyle \int_{1}^{\infty} \frac{1}{\sqrt{e^{2n}-1}}\, dn =\int_{1}^{\infty} (e^{2n}-1)^{1/2} \, dn $ so $u=e^{2n}-1\therefore du=2e^{2n}$

Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...

I have this integral that when solved, involves squares and natural logs, where ##A\,##,##\,b\,##, and ##\,x_e\,## are constants while ##x## is a variable. ##A = \int_{x_e}^{x} \frac{x^2 - b^2}{x} dx = \int_{x_e}^{x} x \, dx -b^2 \int_{x_e}^{x} \frac{dx}{x} = \frac{x^2}{2} - \frac{x_e^2}{2} -...

Homework Statement Evaluate the indefinite integral as a power series. What is the radius of convergence (R)? ##\int x^2ln(1+x) \, dx## Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1## Homework Equations Geometric series ##\frac {1} {1-x} =...

Homework Statement Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of the square is a and one...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

Homework Statement Let ##R## be an integral domain and ##p_1(x),p_2(x) \in R[x]## with neither equal to ##0##. Show that the degree of ##p_1(x)p_2(x)## is the sum of the degrees of ##p_1(x)## and ##p_2(x)##. Homework EquationsThe Attempt at a Solution Here is my attempt. Let ##p_1(x) = a_n...

Homework Statement Calculate the integrals of the following functions on the given paths. Why does the choice of path change/not change each of the results? (c) f(z) = exp(z) / z(z − 3) https://www.physicsforums.com/file:///page1image10808 i. a circle of radius 4 centred at 0. ii. a circle...

Homework Statement Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving:: Homework Equations 3. The Attempt at a Solution [/B] As no...

I used wolframalpha to calculate an integral and here's what I got: ## \displaystyle \int \frac {dy}{y^d} \left( \frac 1 {\sqrt{1-y^{2d}}}-1\right)=\frac{y^{1-d}\left[ y^{2d} \ _2F_1\left( \frac 1 2,\frac{d+1}{2d};\frac {3d+1}{2d};y^{2d} \right)+(d+1)\left( \sqrt{1-y^{2d}}-1 \right)...

This is a chemically inspired problem, but the path is fully quantum mechanics and a bunch of integrals. How does one calculate fully quantum mechanical rate ($\kappa$) in the golden-rule approximation for two linear potential energy surfaces? Attempt: Miller (83) proposes...

Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...

Homework Statement ##\displaystyle \int_{- \infty}^{\infty} \frac{1}{\sqrt{2 \pi}} x e^{- \frac{x^2}{2}} dx## Homework EquationsThe Attempt at a Solution So first off, obviously the answer is 0, because the integrand is odd and we have symmetrical limits of integration. However, when I make...

Does anyone know how I can prove the following equation? ##\displaystyle \frac 1 {d-1}-\int_0^1 \frac{dy}{y^d} \left( \frac 1 {\sqrt{1-y^{2d}} }-1\right)=-\frac{\sqrt \pi \ \Gamma(\frac{1-d}{2d})}{2d \ \Gamma(\frac 1 {2d})} ## Thanks

Homework Statement Calculate the indefinite integral of the function ## \int\frac{3x^3}{\sqrt{1-x^2}}## my book gives the answer ##-(2+x^2)\sqrt{1-x^2}+C## Homework EquationsThe Attempt at a Solution So I started trying to calculate this indefinite integral by using a substitution...

Say we have the following result: ##\displaystyle \int_0^{\infty} \frac{\log (x)}{1 - bx + x^2} = 0##. We see that the denominator is 0 for some positive real number when ##b \ge 2##. Thus, we obtain a two singularities under that condition. Here's my question. Can we go ahead and say that the...

What is the transformation used Is there any explanation for : $$ \frac{\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{t0}\limits^{t}{{\mathrm{(}}{t}\mathrm{{-}}{s}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}}{x}_{0}\mathrm{(}s\mathrm{)}ds $$ How did become like this...

Say we have the following integral: ##\displaystyle \int_0^1 \frac{\log (x+1)}{x^2+1}##. I know how to do this integral with a tangent substitution. However, I saw another method, which was by differentiating ##f## under the integral with respect to the parameter ##t##, where we let...

Just a couple questions. Problem 2: Just would like to know if this is the correct approach for this problem. Problem 3: I am just wondering if I can use Problem 2 to prove the first part of Problem 3? Because to me, they seem very similar. Problem 4: Would I use the MVT for integrals...

I have no idea how to incorporate the limit into the basic definitions for a Riemann integral? All we have learned so far is how to define a Riemann integral and the properties of Riemann integrals. What should I be using for this?

Homework Statement This has been driving me crazy I can't for the life of me figure out how to convert the limits of this integral into spherical coordinates because there is an absolute value in the limits and I'm absolutely clueless as to what to do with with.Homework Equations $$\int_{\frac...

This is a heat equation related math problem. 1. Homework Statement The complete question is: Verify the orthogonality integral by direct integration. It will be necessary to use the equation that defines the λ_n: κ*λ_n*cos(λ_n*a) + h*sin(λ_n*a)=0. Homework Equations κ*λ_n*cos(λ_n*a) +...

Hi the parameter $\lambda$ in linear integral equations say Fredholm that appears in front of the integral containing the kernel i.e. $y(x)=f(x)+ \lambda \int_{a}^{b} \,k(x,t) y(t) dt $ can $\lambda$ be adjusted for the convergence of the method like in using Picard's successive...

∫x2√(3+2x-x2) dx Here's what I've already done: completed the square ∫x2√(4-(x-1)2) dx (x-1) = 2sinθ sinθ = (x-1)/2 x = 2sinθ+1 dx = 2cosθ dθ trig sub + pulled out constants 4∫(2sinθ+1)2√(1-sin2θ)cosθ dθ trig identity 4∫(2sinθ+1)2√(cos2θ)cosθ dθ 4∫(2sinθ+1)2(cos2θ)dθ expanded + trig...

Homework Statement Evaluate ##\displaystyle \int_{1}^{\infty} \frac{dx}{(x+a)\sqrt{x-1}}## Homework EquationsThe Attempt at a Solution First I make the substitution ##u = \sqrt{x-1}##, which ends up giving me ##\displaystyle \int_{0}^{\infty} \frac{2u}{u(u^2 + 1 + a)}du##. Here is where I am...

Homework Statement Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ The Attempt at a Solution If the interpretation of an integral is the derivative at that point, x1 for example, then these can't all be solutions since the tangent, the derivatives...

Homework Statement Hi. Can anyone here solve this integral for me: ##\int\frac{d^3y}{\operatorname dx^3}\;y\;dx## Homework Equations I haven't seen such integral before, it is included in an old math book and it should be solved using "Integration by Parts" technique because it is an exercise...

Homework Statement Let ##R## be a principal ideal domain and suppose ##I_1,I_2,...## are ideals of ##R## with ## I_1 \subseteq I_2 \subseteq I_3 \subseteq ...## The Question has two parts: 1. to show that ##\cup _{i=0}^{\infty}I_i## is an ideal. 2. to show that any ascending as above must...

Homework Statement f(xy)=49/8*e^(−3.5*y) 0 < y < inf and −y < x < y 0 otherwise a. Find the marginal probability density function of X, fX(x). Enter a formula in the first box, and a number for the second and the third box corresponding to the range of x. Use * for multiplication, / for...

Homework Statement Suggest an integral that is reduced to a rational function integral when this substitution is used: ##a)## ##t=\sin x## ##b)## ##t=\sqrt[6] {x+5}## ##c## ##\sqrt{1-9x^2}=-1+xt## Homework Equations 3. The Attempt at a Solution [/B] I found this to be a very interesting...

Homework Statement Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ $$F=ma,~a=\frac{dv}{dt}$$ The Attempt at a Solution $$-\frac{mgR}{s^2}=mv\left( \frac{dv}{ds} \right)\rightarrow~v^2=\frac{2gR}{s}+C$$ $$v_0(s=R)=\sqrt{2gR}\rightarrow~C=2g(R-1)$$...

Homework Statement I'm trying to find the value of the integration: Homework Equations Mod note: Edited the TeX to render properly at this site. Here is the integration written using MAketex: ##\frac{.5}{\sqrt{\pi}}\int_0^{\infty}\exp(-z(1+1.5/v))z^{L-.5} \frac{1}{(\frac{z}{v}+1)^n...

stevendaryl submitted a new PF Insights post Solve Integrals Involving Tangent and Secant with This One Weird Trick Continue reading the Original PF Insights Post.

Homework Statement Hello, I've recently encountered this double integral $$\int_0^1 dv \int_0^1 dw \frac{(vw)^n(1-v)^m}{(1-vw)^\alpha} $$ with ## \Re(n),\Re(m) \geq 0## and ##\alpha = 1,2,3##. Homework Equations I use Table of Integrals, Series and Products by Gradshteyn & Ryzhik as a...

Homework Statement Find f(x) if:(a∈R) $$f(x)=\int_0^{\frac{\pi}{a}}f(t)cos(at-2ax)dt+1$$ Homework Equations $$f(x)=\int_{a}^{b}f(t)dt=F(a)-F(b)$$ $$\int{udv}=uv-\int{vdu}$$ The Attempt at a Solution I tried to use the fundamental of calculus and integration by parts but they don't have answer...