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Svyatoslav
  • Member for 1 year, 6 months
  • Last seen this week
  • Moscow, Россия
13 votes

Can we prove AM-GM Inequality using these integrals?

12 votes
Accepted

Integral of weird function

10 votes

How to prove $\int_0^\infty\frac {\tanh(x)-x\exp(-x)}{x^2}dx=\frac{\zeta'(0)}{\zeta(0)}-\frac{\zeta'(2)}{\zeta(2)}+\gamma-\frac73\log(2)$?

10 votes

A tough integral $\int_0^{\infty}\frac{\operatorname{sech}(\pi x)}{1+4x^2}\, \mathrm dx $

9 votes
Accepted

Finding a closed form for $\sum_{r=0}^{n}{\frac{1}{4^r}\binom{2r}{r}}$

8 votes

How do I solve the double summation $ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{m^2 - n^2}{(m^2 + n^2)^2}$?

8 votes
Accepted

Asymptotic integration of $\frac{ x^{a-\frac{1}{2}} \cos \left(\frac{\pi a}{2}-\alpha x\right)}{(e^x-1)\sqrt{\alpha x}}$

7 votes

Evaluate an absolute monster integral $\int\limits_{0}^{1} \frac{\log(1-x+x^2)}{\sqrt{x}(1+x)}\mathrm{d}x.$

7 votes
Accepted

Trigonometric and exponential integral $\int _0^{\pi }\frac{\cos \left(a\sin x\right)}{1+a\cos x}e^{a\cos x}dx$

6 votes

Sigma and Series involving factorials

6 votes

Compute $\int_0^1 \frac{\sqrt{t(1-t)}}{a+(t-b)^2} \ dt$ for $a,b>0$

6 votes
Accepted

Standard way to evaluate $\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!}$?

6 votes
Accepted

definite integral $\int_0^1 \frac {1-x}{(1+x^3)\ln x}dx=\frac{-1}{2}\ln 3$

5 votes

Is it possible to evaluate the integral $I=\int_0^\infty \frac{\sqrt{x}\arctan(x)}{1+x^2}dx$ with residue theorem?

5 votes
Accepted

Integral $\int_{-\infty}^{\infty}\frac{x^2\cos(x)}{x^4-1}dx$

5 votes

Evaluate $I=\int_{0}^{1}\frac{x^2-x}{(x+1)\ln{x}}dx$

5 votes
Accepted

How to find: $\lim\limits _{n \rightarrow \infty} n^{3} \int\limits_{-\pi}^{\pi} \frac{\cos (2 n x)}{1+x^{8}} d x$

5 votes

Evaluate $\lim_{n\to\infty}\int_1^n\frac{\ln x}{c_n+x\ln x}\,dx$

5 votes

On the integral $\int_0^\infty{\frac{\sin(\lambda x)\mathrm{d}x}{e^{2\pi x}-1}}$

5 votes
Accepted

Is there a closed form for $\lim_{N\to\infty}\left(2\sqrt{N+1}\;-\;\sum_{n=1}^N\frac1{\sqrt{n}}\right)\;$?

5 votes

How do you handle the integral $\int_{0}^{\infty} \frac{d x}{\left(x^{m}+\frac{1}{x^{m}}\right)^{n}}$, where $n\in N?$

4 votes
Accepted

Asymptotics of $\sum_{i=0}^n e^{-i^2/n + i^a}$ as $n \to \infty$

4 votes
Accepted

Fourier transform of $\frac{1}{x^2-a^2} \quad a\in \mathbb{R}$

4 votes

Evaluate the integral $\int\limits_{-\infty}^{\infty} \frac{e^{ax}}{1 + e^x}dx\;$ where $\;0<a<1\;.$

4 votes

Integrate $\int_{0}^{\infty} \big( |y + x|^{\lambda - 1} - |y|^{\lambda - 1} \big) y^{-\beta} dy$

4 votes

$\int_0^1\frac{\ln x\ln^2(1-x^2)}{\sqrt{1-x^2}}dx=\frac{\pi}{2}\zeta(3)-2\pi\ln^32$

4 votes
Accepted

Calculating the integral using residues.

4 votes
Accepted

Fourier transform of $\frac{x}{\sinh(x)}$

4 votes

How to derive Hankel transform of $\frac{1}{\sqrt{r^2+a^2}}$

4 votes
Accepted

Upper bound of $\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(1+x)(1+2x)(1+3x)\cdots(1+nx)} $

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