Articles of integration

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.

Show that $\int_0^1 f^2(x)dx\geq 3$, if $f:\to \mathbb{R}$ is an integrable function s.t. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$

So far I’ve done this, but I don’t know if it will help. Let $\int_0^xf(t)dt=F(x)$. Now, $1=\int xf(x)=F(x)x|_0^1-\int_0^1F(x)dx=1-\int_0^1 F(x)dx$. Then, $\int_0^1 F(x)dx=0$. But I don’t know how to keep going.

Need help with an integral

need help with calculating this: $$\int_{0}^{2\pi}\frac{1-x\cos\phi}{(1+x^2-2x\cos\phi)^{\frac{3}{2}}}d\phi$$ Thanks in advance!

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\,\right)^{s}\, {\Gamma\left(s\right) \over \Gamma\left(s + 2\right)}\,{\rm d}s $$ where β, σ and x are real number I know that Cauchy’s […]

How do I integrate $x^{\frac{3}{2}}e^{-x}$ from 0 to inf?

I have to evaluate the following expression : $$\int^{\infty}_{0} x^{\frac{3}{2}}e^{-x}$$ Wolfram|Alpha evaluates to $\frac{3\sqrt{\pi}}{4}$. I don’t see how we got there. A hint would be helpful. My attempts were to use the “By Parts” rule, when I realized that this is the famous Gamma function. There are several sources on internet which give a way […]

Convergence of improper double integral.

Please help me to determine $\alpha$ and $p$, such that the integral $$ I = \iint_G \frac{1}{(x^{\alpha}+y^3)^p} \ dx dy $$ converges, where $G = {x>0, y >0, x+y <1}$ and $\alpha >0, p>0$. I am comfortable with proper double integrals. I am also comfortable with improper double integrals when $f$ is continuous in $G$ […]