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Svyatoslav's user avatar
Svyatoslav's user avatar
Svyatoslav's user avatar
Svyatoslav
  • Member for 3 years, 4 months
  • Last seen this week
  • Novosibirsk, Россия
25 votes
Accepted

$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\beta} d\alpha d\beta=(0.9999999913...)(\pi/2)$? Seriously?

22 votes

Frullani like Trig integral

19 votes

Conjecture: The sequence $\frac{2}{n} \sum_{i=1}^n \sqrt{\frac{n}{i-\frac{1}{2}}-1}$ converges to $\pi$

15 votes
Accepted

Using complex analysis, show that $\int_0^\infty\frac{\arctan(x)}{1+x^2}\,dx=\frac{\pi^2}8$

14 votes

How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?

13 votes

Can we prove AM-GM Inequality using these integrals?

12 votes
Accepted

Integral of weird function

12 votes

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

12 votes

Compute the integral: $\int_{0}^{\frac{\pi}{2}}\ln(\sec(x)+\tan(x))\csc(x)dx$

11 votes
Accepted

Showing $\int_{-1}^{1} \frac{125}{12}\sqrt[10]{\frac{1 + x}{1 - x}} (x^2 - x) \, dx = \phi\pi$

11 votes
Accepted

Limit of an integral (coming from random walks and networks) as $n\to\infty$

10 votes

Leading order asymptotic behaviour of the integral $\int^1_0 \cos(xt^3)\tan(t)dt$

10 votes

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

9 votes

Evaluating $\int^\infty_0 \frac{\tanh(x)}{x\cosh(2x)}dx$

9 votes

Calculate $\sum_{n=2}^{\infty}\left (n^2 \ln (1-\frac{1}{n^2})+1\right)$

9 votes
Accepted

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

9 votes
Accepted

How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$

8 votes
Accepted

Asymptotic behaviour of an integral with power and exponential functions

8 votes
Accepted

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

8 votes
Accepted

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

8 votes

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

8 votes

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

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

Integral involving $\sin({\ln{x}})$

7 votes

Evaluating $\int_0^\pi x\frac{\sin{\frac{x}{2}} - \cos{\frac{x}{2}}}{\sqrt{\sin{x}}} dx$

7 votes
Accepted

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

7 votes
Accepted

Trouble understanding Blagouchine's extensions to the Malmsten integral

7 votes

Intuition for why $\int_0^\infty\frac{1-(1+x^2)^{-a/2}}{x^2}\,dx$ should equal $\int_0^\infty \frac{1-|\cos x|^a}{x^2} \, dx$?

7 votes

Asymptotic behavior of a sequence of integrals of non-analytic functions

7 votes
Accepted

Evaluating $ \sum\limits_{n=1}^{N-1} \frac{(-1)^n}{\sqrt{1-\cos{\frac{2\pi n}{N}}}} $

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