Integral Definition and 1000 Threads

I am working with the Dilogarithm function and am having problems showing the following and was wondering if someone could help: $$ \int_0^1\int_0^y\left(\frac{1}{x-1}\right)\left(\frac{1}{y}\right)dxdy=-\frac{\pi^2}{6} $$ This is what I have so far: Iterating the first level: $$ \begin{align*}...

##y=x^2ln x-x## ##\frac {dy}{dx}=2x ln x+x-1## ##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x## it follows that, ##\int [2x ln x +x]\,dx##=##x^2 ln x## ##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx## ##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##

(a) i sketched a quarter of a sphere centred at x=0 , y=2 , z=0 (b ) ∫ ∫ √ (4-x2 - (y-2)2) dx dy with limits 0 < x < 2 and 0 < y <4 (c ) i converted to spherical polars and took the integrand as 1/r2 . the volume element is r2sinθ drdθd∅ This leads to the triple integral of sinθ with...

I have some variables that are uncertain, these are w_m = u.ufloat(0.1430, 0.0011) z_rec = u.ufloat(1089.92, 0.25) theta_srec = u.ufloat(0.0104110, 0.0000031) r_srec = u.ufloat(144.43, 0.26) and some constant values c = 299792.458 # speed of light in [km/s] N_eff = 3.046 w_r = 2.469 *...

I have an integral that depends on two parameters ##a\pm\delta a## and ##b\pm \delta b##. I am doing this integral numerically and no python function can calculate the integral with uncertainties. So I have calculated the integral for each min, max values of a and b. As a result I have obtained...

Goodd day, I have a question regarding an exercice I have already posted Bvu was very nice and provided this darwing I already have the solution But y question is : can we use the disk method? because as you can see even though the intersection was at x=-1 the sphere goes deep into the...

Good day here is the solution J just don't understand why the solution r=√2 has been omitted?? many thanks in advance best regards!

I have a integral with unknown h. My integral looks like this where C, a, b are constants F(x) and G(x) are two functions. So the only unknows in the integral is h. How can I solve it ? I guess I need to use scipy but I don't know how to implement or use which functions. Thanks

I cannot understand what this integral is doing: $$g(x)=\left(\frac{i \pi}{2}-\gamma\right) f(x)+\frac{1}{2}\,\text{P.V.}\int_{-\infty}^\infty \left(\frac{1}{x-x'}-\frac{1}{| x-x'| }\right)\,f(x')\,dx'$$ Can anybody please rewrite it in a more understandable form?

Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?

Summary:: Calculate a double integral via appropriate change of variables in R^2 Suppose I have f(x,y)=sqrt(y^12 + 1). I need to integrate y from (x)^(1/11) to 1 and x from 0 to 1. The inner integral is in y and outer in x. How do I calculate integration(f(x,y)dxdy) ? My Approach: I know that...

Writing down several terms of the summation and then doing some simplifying, I get: $$\sum_{r=1}^n \frac{1}{n} \left(1+\frac{r}{n} \right)^{-1}= \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...\frac{1}{2n}$$ How to change this into integral form? Thanks

Summary:: Using an integral and taylor series to prove the Basel Problem The Basel problem is a famous math problem. It asked, 'What is the sum of 1/n^2 from n=1 to infinity?'. The solution is pi^2/6. Most proofs are somewhat convoluted. I'm attempting to solve it using calculus. I notice...

Good Morning To cut the chase, what is the dx in an integral? I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx That said, I have seen it in an integral, specifically for calculating work. I do understand the idea of...

Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...

Hey! :giggle: Let $\displaystyle{I_n(f)=\sum_{i=0}^na_if(x_i)}$ be a quadrature formula for the approximate calculation of the integral $I(f)=\int_a^bf(x)\, dx$. Show that a polynomial $p$ of degree $2n+2$ exists such that $I_n(p)\neq I(p)$. Calculate the approximation of the integral...

Given : The quadratic equation ##x^2+px+q = 0## with coefficients ##p,q \in \mathbb{Z}##, that is positive or negative integers. Also the roots of the equation ##\alpha, \beta \in \mathbb{Q}##, that is they are rational numbers. To prove that ##\boxed{\alpha,\beta \in \mathbb{Z}}##, i.e. the...

Dear all, Last semester on the final exam, our professor gave us an integral that seems difficult to solve. The integral came at the end of a lengthy problem, where we were asked to find the net Gauss curvature of Enneper's surface. The integral that emerged is the following. We tried...

Hi, I'd like to integrate this function: $$ \int _0^ {\pi/2} 2 \sin(x) \cos(x) \sqrt {1+\sin^{2}(x) } dx $$. I think I should introduce some substitution but I'm not sure. How should I proceed?

Hello everyone, in a solution to my measure theory assignment, I have seen the equation $$ \int_{\mathbb{R}}^{} \frac {1}{|x|}\, d\lambda(x)=\infty $$ with ##\lambda## as the 1⁻dim Lebesgue measure. I was wondering how that integral was evaluated as we had never proven any theorem that states...

I've always been taught that the indefinite integral of ##\frac{1}{x}## is ##\ln(|x|)##. Extending this to definite integrals, particularly over limits involving negative values, should work just like any other integral: $$\int_{-1}^{1} \frac {1} {x} dx = \ln(|-1|) - \ln(|1|) = \ln(1) - \ln(1)...

Problem: The sphere is parametrized in cylindrical coordinates by: x = r cosθ y = r sinθ z = (1-r^2)^1/2 and intersected by the cone (x-1)^2 +y^2 = z^2. find the area of the sphere enclosed by the cone using the equation: da = r/(1-r^2) dr dθ Attempt at solution: from the equations for the...

Hello folks, I'm working on a question as follows: I appreciate that there might be more sophisticated ways to do things, but I just want to check that my approach to the line integral is accurate. I will just give my working for the first side of the path. So I have set up the path as a...

The exact value of $$\int_{-1}^1 \arcsin (x) \arccos (x) \arctan(x) \mathrm{d} x.$$

Where does the volume even come from? Any help would be appreciated!

Hello. I need help in simplifying this integral equation, i know the final result must be 2(1-x)^1/2 + C. I been stuck on this one for a while.

Hey! 😊 Calculate using the Simpson's Rule the integral $\int_0^1\sqrt{1+x^4}\, dx$ approximately such that the error is less that $0,5\cdot 10^{-3}$. Which has to be $h$ ? So we use here the composite Simpson's rule, right? An upper bound of the error of that rule is defined as...

I am trying to prove the following expression below: $$ \int _{0}^{1}p_{l}(x)dx=\frac{p_{l-1}(0)}{l+1} \quad \text{for }l \geq 1 $$ The first thing I did was use the following relation: $$lp_l(x)+p'_{l-1}-xp_l(x)=0$$ Substituting in integral I get: $$\frac{1}{l}\left[ \int_0^1 xp'_l(x)dx...

Let $f:[1,\,13]\rightarrow R$ be a convex and integrable function. Prove that $\displaystyle \int_1^3 f(x)dx+\int_{11}^{13} f(x)dx\ge \int_5^9 f(x)dx$,

##\int_0^5 [-x^3+3x^2+6x-8\,]dx## ##\int_0^1 [-x^3+3x^2+6x-8\,]dx= |-\frac {17}{4}|## ##\int_1^4 [ -x^3+3x^2+6x-8\,]dx= 16## ##\int_4^5[-x^3+3x^2+6x-8\,]dx= |-\frac {49}{4}|## Therefore, total area is ##|-\frac {17}{4}|+ 16+|-\frac {49}{4}|=32.5## now where my problem is,... my colleague...

Sometimes I would like to transform an integral ##F(x) = \int_{a}^{x}f(s)ds## into an infinite integral of the form ##F(x) = \int_{0}^{\infty}f(g(u),x)du##. Is there some kind of change of variables that can guarantee this conversion on the boundaries and still give me a function of ##x##, at...

Hello everyone, I have to find an interval of this Riemann integral. Does anybody know the easiest way how to do it? I think we need to do something with denominator, enlarge it somehow. My another guess is the integral is always larger than 0 (A=0) because the whole function is still larger...

Some popular math videos point out that, for example, the value of -1/12 for the divergent sum 1 + 2 + 3 + 4 ... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on. Is there an elementary way to connect this with the more...

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.

this method of derivation is approximating the function using a polyhedron. concentrating on one of the surfaces(say the L'th surface which has an area ##\Delta S_l## and let ##(x_l,y_l,z_l)## be the coordinate of the point at which the face is tangent to the surface and let ##\hat n## be the...

Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...

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...

I'm trying to derive the path integrals, but this step got me confused: Consider the propagator $$K_{q_{j+1},q_j}=\langle q_{j+1}|e^{-iH\delta t}|q_j\rangle $$ Knowing that ##\delta t## is small, we can expand it as $$K_{q_{j+1},q_j}=\langle q_{j+1}|(1-iH\delta t-\frac 1 2 H² \delta...

Hi, I am trying to calculate the heat flow across the boundary of a solid cylinder. The cylinder is described by x^2 + y^2 ≤ 1, 1 ≤ z ≤ 4. The temperature at point (x,y,z) in a region containing the cylinder is T(x,y,z) = (x^2 + y^2)z. The thermal conductivity of the cylinder is 55. The...

I'm a bit confused on the derivation above. I understand what the goal of the derivation is, as it derives Gauss's Law using the solid angle, but i was wondering if someone could kind of fill in the steps the author skipped and explain the use of the solid angle.

Can someone help me with this? Not sure where to start. Exercise 1 (integration) Evaluate the integral ∞ ∫0.2e^−0.2u du. 10

Hi, I apologise as I know I have made similar posts to this in the past and I thought I finally understood it. However, this solution seems to disagree on a technicality. I know the answer ends up as 0, but I still want to understand this from a conceptual point. Question: Evaluate the line...

Hi, This is the first time I see this kind of integral. I'm not sure how to resolve it. ## \int_0^1 F \cdot dr ## ## \int_0^1 F_x(x,0)dx + \int_0^1 F_y(1,y)dy ## ##F = (y,2x)## I don't know the values of ## F_x(x,0) ## and ## F_y(1,y)##

hi guys i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series : $$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$ to evaluate the Gaussian integral as its series some how slimier ...

Hello everyone, I have a maths question (for a change). In summary, I would like to reconcile the following two integrals: Integral A: https://www.wolframalpha.com/input/?i=integrate+(a^2tan^2theta)/(a-b+cos+theta)+dtheta \int\frac{x^2\,dx}{\sqrt{x^2+a^2}(\sqrt{x^2+a^2}-b)} =x...

Hi, I'm trying to solve this integral and then isolate V, but I can't get the right answer. I don't know where is my errors. I probably muffed the integral. ##-bv -cv² = m\frac {dv}{dt}## ## \int_0^t dt = - m \int_{Vo}^v \frac {dv}{bv+cv^2} ## I get this after the integration ##t =...

I was told this problem could simply be solved with calc-1 techniques, so I'm tempted to say we could do d/dx(x∫(limits: 0,x) sin(t) dt. Then it's a simple product rule: d/dx (x) * ∫(0,x) sint dt + x * d/dx(∫ (0,x) sin (t) dt) = 1 - cos(x) + x*sin(x). However, I wonder if we have to allow that...

Hi, I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result. Method: Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...

I was trying to calculate an integral of form: $$\int_{-\infty}^\infty dx \frac{e^{iax}}{x^2}$$ using contour integration, with ##a>0## above. So I would calculate a contour integral with contour being a semicircle that goes along the real axis, closing it in positive direction in the upper...