I have $$\int_{d-1}^{3}\textrm{d}x\left(3-x\right)^3 \sqrt{\left(\frac{2(x-1)}{x}\right) \left(x-\left(d-1\right)\right)}$$ but despite this looking like a simple integral involving fractional powers of $x$ with shifts, Mathematica fails, despite restricting $d$ $\in[3,4]$ and $$\left(\frac{2(x-1)}{x}\right) \left(x-\left(d-1\right)\right)>0$$ Can anyone help with the integral? Why is Mathematica failing?

Let $R$ a region defined by the interior of the circle $x^2+y^2=1$ and the exterior of the circle $x^2+y^2=2y$ and $x\geq 0$, $y\geq 0$ Using polar coordinates $x=r\cos t$, $y=r\sin t$ to determine the region $D$ in $rt$ plane that corresponds to $R$ under this change of coordinate system (polar coordinates) i.e. since $T(r,t)=(r\cos t, […]

I have the following in my notes, but I can’t remember how it works. Please help! $\nabla^2\psi=0, \quad\psi\to 0\quad\text{as}\quad x^2+y^2\to\infty, \quad\psi (x,y,0)$ is continuous Then by using Green’s function, we get the solution to be $$\psi(x',y',z')={z'\over 2\pi}\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty [(x-x')^2+(y-y')^2+z'^2]^{-3\over 2}\psi(x,y,0)\,\,\,dxdy\;.$$ (This part I am sure about.) The primed $x',y',z'$ are the variables introduced when using the […]

I have a differential equation: $$ \frac{dy}{dx} = y \log(y)\cot(x)$$ I’m trying solve that equation by separating variables and dividing by $y\log(y)$: $$ dy = y \log(y) \cot(x) dx$$ $$ \frac{dy}{y \log(y)} = \cot(x) dx$$ $$ \cot(x) – \frac{dy}{y \log(y)} = 0 $$ Where of course $ y > 0 $ regarding to division Beacuse: […]

To go from the definition of the Riemann integral ($f$ is Riemann integrable on $[a,b]$ if there exists a real $A$ such that $\forall \epsilon >0, \exists \delta>0$ such that $\forall D=\{([a_i,a_{i+1}],x_i)\}$ tagged partition of $[a,b]$ $h_i=a_{i+1}-a_i < \delta \implies |S_D(f) – A| < \epsilon$, where $S_D(f)$ is the Riemann sum on $D$) to the […]

So I’m working on washers and I was given the equation of $$1/\sqrt{1+x^2}$$ and I am supposed to rotate the solid around the $x$-axis on the interval of $[-1,1]$. I know that I am supposed to use washers, but I can’t figure out how to find the equation for the outer radius and the inner […]

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} \int_{\mathbb R} |f(x/r)| dx = r^{-1}\int_{\mathbb R} |f(y)| r dy= \int_{\mathbb R} |f(x)| dx;$ and, $\hat{f_{r}} (\xi)= \int_{\mathbb R} f_{r}(x) e^{-2\pi i \xi \cdot x} dx= \hat{f}(r\xi); (\xi […]

Let $l\ge 0$ be an integer. Using integration by parts and the definition of the polylogarithm we have found the following identity: \begin{equation} \int \frac{[\log(\xi)]^l}{(1+\xi)^2} \log(1+\xi) d \xi = \sum\limits_{p=0}^l l_{(p)} \cdot [\log(\xi)]^{l-p} \cdot \phi_p(\xi) \cdot (-1)^{p+1} \end{equation} where the functions $\phi_p$ are given as follows: \begin{eqnarray} \phi_0(\xi) &:=& \frac{\log (\xi +1)+1}{\xi +1} \\ \phi_1(\xi) […]

Any one who knows the conditions for Uniqueness of the Solution to Fredholm’s Integral Equations of the First Kind? Thanks.

Suppose that I have a functional $S[q]=\int_a^b L(t,q(t),q'(t))\,dt$. Such a functional is well-known to extremized by a choice of $q(t)$ satisfying the Euler-Lagrange equation $\dfrac{d}{dt}\dfrac{\partial L}{\partial q’} = \dfrac{\partial L}{\partial q}$. If the integrand has moreover no explicit $t$-dependence, then one can show that the Beltrami identity holds: $L-q’\dfrac{\partial L}{\partial q’}=C$ for some constant $C$. […]

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