Integral Definition and 1000 Threads

How to show that, $$8\pi\int_{0}^{\pi/2}\cos^2x\cdot{\ln^2(\tan^2 x)\over [\pi^2+\ln^2(\tan^2 x)]^2}\mathrm dx=\ln 2-{1\over 4}\zeta(2)$$

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!

$$\int_{0}^{\pi\over 2}{\ln (2\cos x)\over x^2+\ln^2(2\cos x)}\mathrm dx={\pi\over 4}$$ I am very surprise it has a very simple closed form. I was expecting something else, probably involving a few other constants, like $\ln a$, $\pi$ ... I don't know how to go about to show that it is...

I have to deal with this integral in my work, $$\int_{0}^{\infty} \frac{ k^4 e^{-2F^2k^2} }{ (k-k_0)^2 }dk$$ where ##F^2>0 , k_0>0## Is important to mention that it has a double pole in ##k_0## and as a consequence mathematically doesn’t converge. However I have seen before some...

Hi all, I am new to the Maplesoft software and have been experiencing trouble computing numerical integrals. I defined a few mathematical functions in terms of a few variables like so: I then used "subs" to input values to anything that isn't already a defined constant (like ##\hbar,\pi## and...

My first post, and first use of Latex. Here goes. The engineering problem of calculating the time to drain a pipeline, tank, or vessel through an orifice is fairly straightforward using the orifice equation.\(Q=CA_{o}\sqrt{2gh}\) With C being the coefficient of discharge for the orifice, Ao...

Homework Statement Determine if the improper integral is divergent or convergent . Homework Equations - The Attempt at a Solution When i solved the first term using online calculator , the answer was "The integral is divergent" . However , I got 0 . Where is my mistake ?

Homework Statement Determine the area of the surface A of that portion of the paraboloid: [x][/2]+[y][/2] -2z = 0 where [x][/2]+[y][/2]≤ 8 and y≥x Homework Equations Area A = ∫∫ dS The Attempt at a Solution Area A = ∫∫ dS = 3∫∫ dS

Homework Statement I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...

Homework Statement Calculate the line integral ° v ⋅ dr along the curve y = x3 in the xy-plane when -1 ≤ x ≤ 2 and v = xy i + x2 j. Note: Sorry the integral sign doesn't seem to work it just makes a weird dot, looks like a degree sign, ∫.2. The attempt at a solution I have to write something...

Hi, I'm re-studying integrals and I got stuck with this problem. Actually the math beyond it is very clear but I still can figure it out. Take this function: ##f(x) = \begin{cases} 0, & x \lt 1 \\ 1, & x = 1 \\ 0, & x > 1 \end{cases} ## According to Spivak's Calculus I, a function is...

Homework Statement \int_0^{2018 \pi} \lvert \sin(2018x) \lvert \mbox{d}x Homework EquationsThe Attempt at a Solution So the period is: \frac{2 \pi}{ 2018} Each "hump" of the sine has an area of 2 so if I count the number of humps I am done. In one period of an absolute sine function the...

Homework Statement Homework Equations V=kq/x The Attempt at a Solution I know the correct solution. It's... On my first attempt, rather than use (d+x) in the denominator and integrate from 0 to L, I instead used (x) and integrated from (d) to (L+d). This produces the wrong answer...

Hello,I am not sure if these types of problems are Intermediate or advanced. I am not sure too whether they have a certain name or not. I have a function inside a definite integral. The solution of this definite integral is known. What is the function that satisfy the known solution. In...

As shown in the image below, I tried to integrate a large integral. However, the result is strange. According to the result, the integral is always zero whatever the values of w, h, L, P, S and k. However, when I try to put some "test values", the result is not zero. test values...

Here is this week's POTW: ----- If $n$ is a positive integer, evaluate $$\int_{0}^\infty \frac{dx}{1 + x^n}$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Hi, I am trying to solve this integral: \int_0^{\infty}\frac{\ln(1+\alpha\,x)}{(1+x)^2}\,dx Using integration by parts this can be written as: -\frac{1}{1+x}\ln(1+\alpha\,x)\Big|_0^{\infty}\Big. + \alpha\int_0^{\infty}\frac{1}{(1+x)(1+\alpha\,x)}\,dx The first term evaluates to zero. The...

The area of the region y=-x^2+6x, y=x^2-2x, Y-axis, and the line x = 3 is ... A. 16 unit area B. 18 unit area C. \frac{64}{3} unit area D. 64 unit area E. 72 unit area Sorry I couldn't post the graph, but I interpreted it as \int_0^3(-x^2+6x-x^2+2x)dx-\int_0^2(x^2-2x)dx and got \frac{31}{3}...

Hi, I posted a question here a few days ago regarding some questions I've been doing on an online quiz. I seem to be getting stuck on the integral substitution questions. I've been slowly making progress, but some of these questions have been confusing me, and reading up on them is only giving...

Homework Statement I have to prove that the improper integral ∫ ln(x)/(1-x) dx on the interval [0,1] is convergent. Homework Equations I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1. The Attempt at a Solution The function can be approximated to ln(x) when it approaches...

Evaluate $$I = \int_{-2}^{0} \frac{x}{\sqrt{e^x+(2+x)^2}}\,dx$$

what is the integral of 2^(2x)? need help tonight, exam is tomorrow teacher says the answer is: 2^(2x) / 2Ln(2) why 2 times Ln(2)?

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as...

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as follows: At the end of the above proof...

Dear friends: It's strange that Mathematica can do the integral of ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##, however, fails when it's changed to ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##. Are there any major differences between this two types? Is it possible to do the second kind of integral...

Evaluate $\displaystyle \int_C(x+y)ds$ where C is the straight-line segment $x=t, y=(1-t), z=0, $ from (0,1,0) to (1,0,0) ok this is due tuesday but i missed the lecture on it so kinda clueless. i am sure it is a easy one.

$\tiny{232.5a}\\ \textsf{Evaluate the double integral}$ \begin{align*}\displaystyle I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\ R&=[0,2]\times[-1,1] \end{align*} Ok, just want to see if I made the first step correct. this looks like simply a rectangle so x and y are basically...

Homework Statement I have the following integral I wish to solve (preferably analytically): $$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$ where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...

Homework Statement I have an integral $$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$ where j and c are complex, which I'd like to write in terms of ## \text{erf}## I'd like to know what would happen to the integral limits as I make the change of variables ##t = jp - c##. 1) As ##p## tends...

$\textsf{Find the volume of the given solid region bounded by the cone}$ $$\displaystyle z=\sqrt{x^2+y^2}$$ $\textsf{and bounded above by the sphere}$ $$\displaystyle x^2+y^2+z^2=128$$ $\textsf{ using triple integrals}$ \begin{align*}\displaystyle V&=\iiint\limits_{R}p(x,y,z) \, dV...

<Moderator's note: Moved from a technical forum and thus no template.> where a, b, c, d and n, all are positive integers. Find the value of 'c'. ------------------------------- I don't really have a good approach for this one. I just made a substitution u = sinx + cosx I couldn't clear up...

calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)

Homework Statement Homework EquationsThe Attempt at a Solution I think the answer for number 1 , graph somewhat like this I get trouble for 2, 3, etc I (k) = ##\int_{-1}^{1} f(x) dx ## f(x) = ## \mid x^2 - k^2 \mid## 2) k < 1 for negative side ##\int_{-1}^{-k} (x^2 - k^2) dx +...

$\textsf{Reverse the order of integration in the following integral }$ \begin{align*}\displaystyle I&=\int_0^1 \int_2^{2e^x}f(x,y) \quad dy \, dx \end{align*} $\textit{From the integral we have that}$ $$0\leq x\leq 1 \quad \textit{and} \quad 2\leq y\leq 2e^x$$ $\textit{So, we get that}$...

Homework Statement Solve the integral: [/B] ##\int_{-\infty}^\infty {\frac {\cos(x)}{x^2 + 1}} \, dx##Homework EquationsThe Attempt at a Solution I'm a bit stuck here, so what to substitute for tan \theta so as to compute the integral? Help me out here please

Question #1 - A lobster tank in a restaurant is 1 m long by 0.75 m wide by 60 cm deep. Taking the density of water to be 1000 kg/m$^3$, find the water forces (a) on each of the larger sides of the tank; (b) on each of the smaller sides of the tank.

Homework Statement A specific problem of the Fredholm integral equation is given as $$\phi(x) = \pi x^2+\int_0^\pi3(0.5\sin(3x)-tx^2)\phi(t)\,dt$$ and the exact solution is ##\phi(x) = \sin 3x##. Homework Equations Nothing comes to mind. The Attempt at a Solution I'm unsure how to approach...

\begin{align}\displaystyle v_{\tiny{s6.15.6.3}}&=\displaystyle \int_{0}^{1}\int_{0}^{z}\int_{0}^{x+z} 6xz \quad \, dy \, dx\, dz \end{align} $\text{ok i kinda got ? with $x+z$ to do the first step?}\\$ $\text{didn't see an example}$

<Moderator's note: Image substituted by text.> 1. Homework Statement Given the following vector field, $$ \dfrac{2(x-1)\,dy - 2(y+1)\,dx}{(x-1)^2+(y+1)^2} $$ how do I integrate : The integral over the curve x^4 + y^4 = 1 x^4 + y^4 = 11 x^4 + y^4 = 21 x^4 + y^4 = 31 Homework Equations...

Hello all, I am trying to solve the integral: \[\int cot(x)\cdot csc^{2}(x)\cdot dx\] If I use a substitution of u=cot(x), I get \[-\frac{1}{2}cot^{2}(x)+C\] which is the correct answer in the book, however, if I do this: \[\int \frac{cos(x)}{sin^{3}(x)}dx\] I get, using a substitution...

The big blue circle has been put there by my math prof to denote the location of the error in the following solution. Why is this an error? I'm lost. :(

True or False: If $f(x)$ is a negative function that satisfies $f'(x) > 0$ for $0 \le x \le 1$, then the right hand sums always yield an underestimate of $\int_{0}^{1} (f(x))^2\,dx$. - - - Updated - - - Would it be true since right hand Riemann sums for a negative, increasing function will...

True or False: Let $F(x)$ be an antiderivative of a function $f(x)$. Then, $F(2x)$ is an antiderivative of the function $f(2x)$.

True or False: If $$h(t) > 0$$ for $$0 \le t\le 1$$, then the function $$H(x) = \int_{0}^{x} h(t)\,dt$$ is concave up for $$0 \le t\le 1$$.

$\tiny{244 .15.7.24}$ $\textsf{Evaluate the spherical coordinate integral}\\ \begin{align}\displaystyle DV_{24}&=\int_{0}^{3\pi/4} \int_{0}^{\pi} \int_{0}^{1} \, 5\rho^3 \sin^3 \phi \, d\rho \, d\phi \, d\theta \\ &=\int_{4}^{3\pi/4}...