What is the integral of this function : $$\int_{-\infty}^{+\infty}x^{\alpha} \sin{ax} \, \mathrm{d}x$$ with $\alpha, a$:real values?

I was trying to answer a question about a random walk when I came across the integral $$ \int_0^\infty \sum_{m=0}^\infty\frac{(-1)^m}{2m+1}\left(1-e^{-(2m+1)^2/x^2}\right)\mathrm{d}x. $$ For probabilistic reasons, I think it has a finite value. Is there a simple proof of this? Is there a way to compute or simplify the expression? If one could exchange the $\int$ with […]

I was trying to calculate the following integral: $\displaystyle\int_1^\infty\frac{dx}{x \lfloor x \rfloor}=? $ which I found to be equivalent to $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k(k+1)} $. There is a close relative of this sum which is $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k^2}=-\zeta^\prime(2)$ and its value is known in terms of Glashier-Kinkelin constant (A): $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k^2}= \frac{\pi^2}{6}[12\ln(A)-\gamma-\ln(2\pi)]$ My idea for solving this was to assume a […]

I want to solve following integral $$\int_0^t\cdots \int_0^t (x_1+x_2\cdots x_M)dx_1dx_2\cdots dx_M$$ Is there some way through which I could convert these multiple integrals to single integral (for example through change of variable like $y=x_1+x_2\cdots x_M$). Any help in this regard is much appreciated.

This question already has an answer here: Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$ 5 answers

$\int_{0}^{\infty}\frac{(x^{4}+2017)*(sin x)^{3}}{x^{5}+2018*x^{3}+1}dx$ How do I prove that it converges and that it does not converge absolutely?

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}) […]

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: $$c\cdot e_k =\dfrac{1}{V}\int_{\partial P}\dfrac{1}{2}\left(x\cdot e_k\right)^2\left(n\cdot e_k\right)=\dfrac{1}{2V}\sum\limits_{i=0}^{N-1}\int_{A_i}\left(x\cdot e_k\right)^2\left(n_i\cdot e_k\right), k=1,2,3$$ It remains to compute that: $$\int_{A_i}\left(x\cdot e_k\right)^2\left(n_i\cdot e_k\right)=\dfrac{1}{6}{\hat n}\cdot e_k\left(\left[\tfrac{1}{2}\left(a_i+b_i\right)\cdot e_k\right]^2+\left[\tfrac{1}{2}\left(b_i+c_i\right)\cdot e_k\right]^2+\left[\tfrac{1}{2}\left(c_i+a_i\right)\cdot […]

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 […]

Problem: $f'(x)$ is a continuous function for $[0, 1]$. Show that $$\lim_{n \to \infty}n \left( \frac{1}{n}\sum_{i=1}^n {f\left(\frac{i}{n}\right)-\int_0^1f(x)dx} \right)=\frac{f(1)-f(0)}{2}$$ I tried to use the definition of the definite integral to change it to a limit but it doesn’t seem to work. And I also wonder why a continuous condition for $f'(x)$ was given. I thought it […]

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