Articles of improper integrals

Prob. 10 (d), Chap. 6, in Baby Rudin: Holder's Inequality for Improper Integrals

Here is the link to my earlier post here on Math SE on Probs. 10 (a), (b), and (c), Chap. 6, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Probs. 10 (a), (b), and (c), Chap. 6, in Baby Rudin: Holder's Inequality for Integrals Now I’ll be attempting Prob. 10 (d). […]

Convergence of improper integrals and asymptotic behaviour

Is it correct to just consider the asymptotic behaviour of the integrand in an improper integral to determine whether or not it converges? For example, $\frac{1}{(x+3)^2}\sim_{\infty}\frac{1}{x^2}$. Since $\int_1^{\infty}\frac{1}{x^2} dx$ converges, can I conclude that $\int_1^{\infty}\frac{1}{(x+3)^2} dx$ does as well?

Is there a way to compute $\int_0^\infty \frac{\cos (qt) J_1 (qr)}{1+q^2} \, \mathrm{d} q$ provided that $0<t<r$?

In a dual integral situation, the following integral has to be involved $$ \int_0^\infty \frac{\cos (qt) J_1 (qr)}{1+q^2} \, \mathrm{d} q \quad\quad (0<t<r) \, . $$ Visibly this integral is convergent. I was wondering whether an amenable analytical expression is possible? This will be useful for my further analysis. Any help is highly appreciated. Thanks. […]

$\int_{1}^{\infty} h(x)\ dx$ converges $\Rightarrow$ $h$ is bounded in $[1, \infty)$

Let $h:[1, \infty)\rightarrow \mathbb R$ a continuous non-negative function, such that $\int_{1}^{\infty} h(x)\ dx$ converges. does $h$ must be bounded in $[1, \infty)$? I tried to prove it by showing that if $\int_{1}^{\infty} h(x)\ dx$ converges, then by the definition the $\lim_{b \to \infty}\int_{1}^{b} h(x)\ dx$ exists. Can I conclude that from the existence of […]

Example of continuous positive function without limit whose improper integral is convergent

I would like an example of a function that is continuous and positive and has the following properties: $$\int_a^{\infty}f(x) dx $$ is convergent and $$\lim_{x \to \infty} f(x) \not = 0$$ (I think the limit should not exist).

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 […]

Test for convergence

Possible Duplicate: Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral? I am stuck with the following integral: \begin{equation} \int_\mathbb{R} \frac{\sin t}{t} \end{equation} I would like to find out whether this integral is convergent, but I totally forgot how to find the right convergence test here, at the origin I […]

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 $G$ […]

$\int^{\pi/2}_{0}\log|\sin x| \,dx = \int^{\pi/2}_{0}\log|\cos x| \,dx $

Prove that : $$\int^{\pi/2}_0 \log|\sin x| \,dx = \int^{\pi/2}_0 \log|\cos x| \,dx $$ I tried to cut the integral into a sum of parts and changing variable but it didn’t work out right, i dont know how to solve this kind of problems in any other way, any hint will be much appreciated!

Determining whether $\int_{0}^{\infty} \frac{x \sin(x)}{1+x^2}dx$ converges and converges absolutely

I would like to check whether $$\int_{0}^{\infty} \frac{x \sin(x)}{1+x²}dx$$ converges and converges absolutely. I have a feeling that neither is true, however none of the methods known to me seem to help. I struggle to find a lower estimate for the function. Any hints and help welcome. I tried using $$\frac{x \sin(x)}{1+x²}\leq \frac{x \sin(x)}{x²}=\frac{ \sin(x)}{x}$$ […]