Articles of integration

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 $G$ […]

Surface area of ellipsoid given by rotating $\frac{x^2}{2}+y^2=1$ around the x-axis

Calculate the surface area of the ellipsoid that is given by rotating $\frac{x^2}{2}+y^2=1$ around the x-axis. My idea is that if $f(x)=\sqrt{1-\frac{x^2}{2}}$ rotates around the x-axis we will end up with the same figure. The formula for calculating the surface area is: $ 2\pi \int_{-\sqrt{2}}^{\sqrt{2}} f(x)\sqrt{1+f'(x)^2}dx $ and with $f'(x)^2 = \frac{x^2}{4-2x^2}$ the integral becomes […]

Understanding Dirac delta integrals?

I’m confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$ and am told this evaluates to 8. Can anyone please step-by-step how to get there? I’m really struggling to understand how this integration works – I know that $\int \delta (x) =1$ but can’t […]

Region of convergence of $\int_0^{\infty} x^s e^{-\frac{|\log(x)|}{2}}dx$ where $s \in \mathbb{C}$

I am interested in finding the region of convergence of the integral \begin{align} \int_0^{\infty} x^s e^{-\frac{|\log(x)|}{2}}dx \end{align} where $s \in \mathbb{C}.$ How do we approach this type of proablem?

Integrating by parts. A possible typo in the notes from Hairer

In page 6 of the notes “Introduction to Stochastic PDE’s” from Martin Hairer, one reads: Let’s write it down to check i: $$C(t,t,0,x) = \frac{|x|}{8}\int^\infty_{\frac{|x|^2}{8t}} z^{-3/2} e^{-z}\,dz \\ = \frac{|x|}{8}\int^\infty_{\frac{|x|^2}{8t}} (-2 z^{-1/2})’ e^{-z}\,dz \\ = \frac{|x|}{8} (-2 z^{-1/2}) e^{-z}\bigg]^\infty_{\frac{|x|^2}{8t}} – \frac{|x|}{4}\int^\infty_{\frac{|x|^2}{8t}} z^{-1/2} e^{-z}\,dz \\ =\frac{|x|}{8} 2 \bigg(\frac{|x|^2}{8t} \bigg)^{-1/2} e^{-\frac{|x|^2}{8t}} – \frac{|x|}{4}\int^\infty_{\frac{|x|^2}{8t}} z^{-1/2} e^{-z}\,dz \\ =\frac{\sqrt{2t}}{2} […]

$\lim_{x\to\infty}\int_x^{x+1}f(y)dy=0$ implies $\lim_{x\to\infty}\frac{\int_0^{x}f(y)dy}{x}=0$

Let $f$ be a non negative continuous function on $[0,\infty)$ such that $$\lim_{x\to\infty}\int_x^{x+1}f(y)dy=0.$$ How do we prove that $$\lim_{x\to\infty}\frac{\int_0^{x}f(y)dy}{x}=0.$$ If we see this question using the primitive function, do we have the following result for a continuous function $F$: $$\lim_{x\to\infty}F(x+1)-F(x)=0$$ implies that $$\lim_{x\to\infty}\frac{F(x)}{x}=0.$$