User oO_ƲRF_Oo - Mathematics Stack Exchange

14 votes

Accepted

What is the mean value of $|\sin x +\sin (\pi x)|$?

answered Jul 18, 2023 at 7:15

9 votes

Evaluate $\int\limits_0^{\sqrt 2 } {\frac{1}{{3{a^2} + 2}}\frac{{\arctan \left( {\sqrt {{a^2} + 1} } \right)}}{{\sqrt {{a^2} + 1} }}da}$

answered Sep 5, 2023 at 8:19

8 votes

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

answered Feb 10, 2023 at 21:38

7 votes

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

answered Sep 14, 2023 at 21:30

5 votes

Conjecture: $\,\lim\limits_{n\to\infty}\int_0^1 (1+|\sin{nx}|)^{-2}\mathrm dx=\frac{4}{3\pi}$

answered Jul 16, 2023 at 18:47

5 votes

Accepted

Proving that $\int_{0}^{1}\left(\zeta(t)+\frac{1}{1-t}\right)dt=\sum_{n=0}^{\infty}\frac{\gamma_{n}}{(n+1)!}$

answered Nov 25, 2023 at 17:53

5 votes

Accepted

Tough integrals: $\int_0^1\frac{\log x\arctan\left(\frac{\log x}{2\pi}\right)}{1+x^2}dx$

answered Sep 15, 2023 at 11:31

4 votes

Accepted

Compute ; $\Gamma=\lim_{n\to\infty} \int_{t_n}^{t_{n+1}} \frac{(f(x-t_n))^{g(t_{n+1}-x)}}{(f(t_{n+1}-x))^{g(x-t_n)}+(f(x-t_n))^{g(t_{n+1}-x)}} \, dx$

answered Jul 22, 2023 at 2:10

4 votes

Accepted

Indefinite or definite Trigonometric Integral

answered Aug 5, 2023 at 11:14

4 votes

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

answered Sep 21, 2023 at 7:48

3 votes

Evaluate series: $\sum_{k=1}^\infty (-1)^k\left[ k\ln\left(\frac{k^4+2k^3+k^2}{k^4+2k^3+3k^2+2k+2}\right)+\ln\frac{k^2+2k+1}{k^2+1} \right]$

answered Sep 13, 2023 at 12:09

3 votes

To evaluate integral using Beta function

answered Feb 15, 2023 at 14:20

3 votes

Accepted

Integral $\int_{0}^{\pi}\arctan\left(\cot\left(mt\right)\right)\arctan\left(\cot\left(nt\right)\right)dt$

answered Aug 25, 2023 at 16:21

3 votes

Accepted

Limit $\lim_{k\to\infty}\left(\sum_{r=1}^{k-1}\zeta\left(2r\right)\frac{\left(-1\right)^{r+k}}{\left(2k-2r-1\right)!}\right)$

answered Aug 28, 2023 at 6:57

2 votes

Encountered $\displaystyle{\sum_{k=1}^\infty\frac{(-1)^k\sin(kt)}{k}}$ while solving another integral

answered Sep 17, 2023 at 2:21

2 votes

An integral of $\,\int_{0}^{\infty}{\!\frac{\ln\left(1+{\frac{1}{x}}\right)}{π^2+\ln^2x}\mathrm{d}x}$

answered Jul 15, 2023 at 14:24

1 vote

Improper Integral $\int_0^\infty\left(\frac{\tanh(x)}{x^3}-\frac1{x^2\cosh^2(x)}\right)dx = \frac{7\zeta(3)}{\pi^2} $

answered Jan 31, 2023 at 6:55

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17 Answers

What is the mean value of $|\sin x +\sin (\pi x)|$?

Evaluate $\int\limits_0^{\sqrt 2 } {\frac{1}{{3{a^2} + 2}}\frac{{\arctan \left( {\sqrt {{a^2} + 1} } \right)}}{{\sqrt {{a^2} + 1} }}da}$

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

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

Conjecture: $\,\lim\limits_{n\to\infty}\int_0^1 (1+|\sin{nx}|)^{-2}\mathrm dx=\frac{4}{3\pi}$

Proving that $\int_{0}^{1}\left(\zeta(t)+\frac{1}{1-t}\right)dt=\sum_{n=0}^{\infty}\frac{\gamma_{n}}{(n+1)!}$

Tough integrals: $\int_0^1\frac{\log x\arctan\left(\frac{\log x}{2\pi}\right)}{1+x^2}dx$

Compute ; $\Gamma=\lim_{n\to\infty} \int_{t_n}^{t_{n+1}} \frac{(f(x-t_n))^{g(t_{n+1}-x)}}{(f(t_{n+1}-x))^{g(x-t_n)}+(f(x-t_n))^{g(t_{n+1}-x)}} \, dx$

Indefinite or definite Trigonometric Integral

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

Evaluate series: $\sum_{k=1}^\infty (-1)^k\left[ k\ln\left(\frac{k^4+2k^3+k^2}{k^4+2k^3+3k^2+2k+2}\right)+\ln\frac{k^2+2k+1}{k^2+1} \right]$

To evaluate integral using Beta function

Integral $\int_{0}^{\pi}\arctan\left(\cot\left(mt\right)\right)\arctan\left(\cot\left(nt\right)\right)dt$

Limit $\lim_{k\to\infty}\left(\sum_{r=1}^{k-1}\zeta\left(2r\right)\frac{\left(-1\right)^{r+k}}{\left(2k-2r-1\right)!}\right)$

Encountered $\displaystyle{\sum_{k=1}^\infty\frac{(-1)^k\sin(kt)}{k}}$ while solving another integral

An integral of $\,\int_{0}^{\infty}{\!\frac{\ln\left(1+{\frac{1}{x}}\right)}{π^2+\ln^2x}\mathrm{d}x}$

Improper Integral $\int_0^\infty\left(\frac{\tanh(x)}{x^3}-\frac1{x^2\cosh^2(x)}\right)dx = \frac{7\zeta(3)}{\pi^2} $