Articles of integration

Definite Integral of the following term with $e^e$

If anyone can help me solve the following equation I will really appreciate it. It is not part of any assignment or DIY kind of thing. I am trying to solve one research paper and it is part of the bigger problem. Seriously stuck here for two days. $\int^\infty_0 t(n\lambda e^{-ne^{-\lambda t} – \lambda t}) […]

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: $$f_m(x)=\frac{n!}{(k!)^2}2^{-n}\frac{1}{b}e^{-(k+1)|x|/b}(2-e^{-|x|/b})^k$$ where the p.d.f. of the underlying Laplace distribution is given as $f(y)=\frac{1}{2b}e^{-|y|/b}$. The formula for p.d.f. of the median stems from the usual method […]

prove an integral reduction formula for $cos^{m}x/sin^{n}x$

prove that: $$\int {cos^{m}x\over sin^{n}x}dx = {cos^{m-1}x\over (m-n)sin^{n-1}x} + {m-1\over m-n}\int {cos^{m-2}x\over sin^{n}x}dx +C$$ What I did: $$\int cos^{m}xsin^{-n}xdx= \int cosxcos^{m-1}xsin^{-n}xdx$$ then I used integration by parts: $u=cos^{m-1}x$ $$du=-(m-1)cos^{m-2}xsinxdx$$ $$dv=cosxsin^{-n}xdx$$ and $$v={sin^{-n+1}x\over-n+1}$$ hence: $$\int {cos^{m}x\over sin^{n}x}dx= -{cos^{m-1}x\over (n-1)sin^{n-1}x} – {m-1\over n-1}\int {cos^{m-2}x\over sin^{n-2}x}dx$$ So I would like you to tell me what can I do […]

Double integral. Polar coordinates

Change double integral to an equivalent double integral in terms of polar coordinates. $$\int_0^2\int_0^x y\,dy\,dx$$ Firstly: $$x=r\cos\phi$$ $$y=r\sin\phi$$ I am trying to think of bounds for $r$ and $\phi$ but I am having a hard time. My guess that the bound of $\phi$ is $0\leq \phi \leq \frac{\pi}{4}$. Can’t really think of a good bound […]

Integral from ball to sphere

Well I am working something, which deals with the following problem: For example, I want to compute an integral $\int_{B(0,R)}f(x)dx$, where $B(0,R)=\{x\in\mathbb R^n:\;|x|\leq R\}$ and $S(0,R)=\{|x|=R\}$. Now we have the following formula $$ \int_{B(0,R)}f(x)dx=\int_0^Rdr\int_{S(0,r)}f(y)dS(y) $$ My question is: I assume only that $f(x)$ is a measurable respect to Lebesgue measure and non-negative. Does the above […]

How to arrive at Stokes's theorem from Green's theorem?

I would like to verify the identity $$ \oint \vec F \cdot (\hat i dx + \hat j dy) + \oint \vec F \cdot (\hat i dx + \hat j dy) + \oint \vec F \cdot (\hat i dx + \hat j dy) = \oint \vec F \cdot (\hat i dx + \hat j dy […]

Integrate $\log(x)$ with Riemann sum

In a homework problem I am asked to calculate $\int_1^a \log(x) \mathrm dx$ using a Riemann sum. It also says to use $x_k := a^{k/n}$ as steps for the stair functions. So far I have this: My step size is $x_k – x_{k-1}$ which can be reduced to $a^{\frac{k-1}{n}} (a^{\frac{1}{n}} -1)$. The total sum then […]

Change the order of conditional expectation of integration

I encountered this problem when learning SDE: $g(t,\omega)$ is a adapted process then $$\mathbb E\left(\int_a^b |g(t)|^2 \, dt \mid \mathcal F_a\right)=\int_a^b\mathbb E\left(|g(t)|^2\mid \mathcal F_a\right) \, dt$$ I don’t know whether I can change the order of conditional expectation of integration. So I try to prove it using the definition of conditional expectation but failed, does […]

Finite content which is not a pre-measure

I’ve just run into an apparent contradiction and it would be great if someone could explain where I’m going wrong: A basic theorem in measure theory states that for a finite content $\mu$ on a ring of subsets (of some set $X$) one has $\mu$ is a pre-measure if and only if for every sequence […]

$\int \frac{x^{2} \arctan x}{1+x^{2}}dx$

So I have the integral $$\int \frac{x^{2} \arctan x}{1+x^{2}}dx$$ how can I solve this integral without substituting $u=\arctan x$? Because I think that if I do this, lets suppose that in the end I’ll have as a solution $\tan u + \cos u$ when I replace $u$, it will look strange.