User Zacky - Mathematics Stack Exchange

66 votes

1 answer

3k views

Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

asked Dec 28, 2019 at 21:03

45 votes

7 answers

4k views

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

asked Dec 23, 2018 at 20:53

41 votes

6 answers

2k views

Integral for the New Year $2019$!

asked Dec 31, 2018 at 22:00

36 votes

4 answers

3k views

Relation between the roots of a function

asked Feb 21, 2018 at 13:31

25 votes

4 answers

2k views

Integral $\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}$

asked Jan 1, 2019 at 17:30

25 votes

1 answer

985 views

Integral $\int_0^1 \frac{dx}{\prod_{n=1}^\infty (1+x^n)}$

asked Dec 15, 2018 at 10:28

23 votes

1 answer

1k views

Integral $\int_0^1 \frac{2x-1}{1+x-x^2}\left(4\ln x\ln(1+x)-\ln^2(1+x)\right)dx$

asked Oct 5, 2019 at 14:26

21 votes

3 answers

4k views

A lucky proof for the Basel problem.

asked Jul 28, 2019 at 23:51

21 votes

2 answers

2k views

Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$

asked Jan 12, 2019 at 15:41

17 votes

4 answers

838 views

Proving that $\sum_{n=1}^\infty \frac{\sin\left(n\frac{\pi}{3}\right)}{(2n+1)^2}=\frac{G}{\sqrt 3} -\frac{\pi^2}{24}$

asked Oct 20, 2018 at 13:50

17 votes

1 answer

928 views

Integral $\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx$

asked Jun 3, 2019 at 0:16

16 votes

2 answers

2k views

Integral $\int_0^\infty \frac{\arctan(x^2)}{x^4+x^2+1}dx$

asked Nov 3, 2018 at 20:54

16 votes

0 answers

625 views

Odd values for Dirichlet beta function

asked Apr 24, 2018 at 13:40

14 votes

3 answers

1k views

Integral $\int_0^1 \frac{dx}{\sqrt[3]{x(1-x)}(1-x(1-x))}$

asked Jul 5, 2019 at 12:50

13 votes

6 answers

2k views

Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

asked Apr 28, 2019 at 18:00

13 votes

2 answers

622 views

Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

asked Sep 28, 2018 at 23:05

12 votes

1 answer

325 views

Prime pattern found in the integral $\int_0^\pi \exp\left(\frac{\cos t}{8}\right)\cos\left(\frac{\sin t}{8}\right) \cos(nt )dt$

asked Sep 11, 2018 at 11:26

11 votes

1 answer

476 views

Proving that $(-1)^{n+1}G_n =\int_0^\infty \frac{1}{(1+x)^n (\pi^2+\ln^2 x)} dx$

asked Oct 5, 2018 at 14:32

11 votes

3 answers

489 views

Integral $\int_0^{\pi} \frac{\cos(2018x)}{5-4\cos{x}}dx$

asked Jun 15, 2018 at 7:50

11 votes

3 answers

2k views

Integral $\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$

asked Aug 8, 2018 at 23:21

11 votes

2 answers

582 views

Integral $\int^{\frac{\pi}{6}}_{0}\ln^2(2\sin \theta)d\theta$

asked May 10, 2018 at 14:31

10 votes

2 answers

1k views

Integration trick $\int^{2\pi}_{0} f(a+ r \cos \theta, b+r\sin \theta)d\theta=2\pi f(a,b)$

asked Mar 17, 2018 at 12:52

10 votes

2 answers

951 views

Integral $\int_0^{\frac{\pi}{2}} x^2 \sqrt{\sin x} \, dx$

asked Jul 6, 2018 at 19:53

10 votes

5 answers

962 views

Integral $\int_0^1 \frac{(x^2+1)\ln(1+x)}{x^4-x^2+1}dx$

asked Oct 18, 2018 at 21:47

10 votes

2 answers

680 views

Integral $\int_0^1 \frac{\ln(1+x)}{1+x^3}dx$

asked Apr 19, 2019 at 19:37

9 votes

4 answers

3k views

Integral $\int_0^1 \frac{\arctan x}{x^2-x-1}dx$

asked Feb 8, 2019 at 19:41

9 votes

1 answer

159 views

Bounds for $\int_0^1 x^{\cos x+\sin x} dx$

asked May 13, 2018 at 10:56

9 votes

1 answer

358 views

Integral $\int_1^3 \frac{\log x}{x^2+3}dx$

asked May 28, 2018 at 8:43

8 votes

2 answers

442 views

Closed form for $\int_0^1 \frac {\log^n(x)}{(1-x)^m} dx$

asked Apr 12, 2018 at 10:30

8 votes

1 answer

429 views

Integral $\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$

asked Nov 18, 2018 at 18:00

Stack Exchange Network

80 Questions

Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

Integral for the New Year $2019$!

Relation between the roots of a function

Integral $\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}$

Integral $\int_0^1 \frac{dx}{\prod_{n=1}^\infty (1+x^n)}$

Integral $\int_0^1 \frac{2x-1}{1+x-x^2}\left(4\ln x\ln(1+x)-\ln^2(1+x)\right)dx$

A lucky proof for the Basel problem.

Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$

Proving that $\sum_{n=1}^\infty \frac{\sin\left(n\frac{\pi}{3}\right)}{(2n+1)^2}=\frac{G}{\sqrt 3} -\frac{\pi^2}{24}$

Integral $\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx$

Integral $\int_0^\infty \frac{\arctan(x^2)}{x^4+x^2+1}dx$

Odd values for Dirichlet beta function

Integral $\int_0^1 \frac{dx}{\sqrt[3]{x(1-x)}(1-x(1-x))}$

Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

Prime pattern found in the integral $\int_0^\pi \exp\left(\frac{\cos t}{8}\right)\cos\left(\frac{\sin t}{8}\right) \cos(nt )dt$

Proving that $(-1)^{n+1}G_n =\int_0^\infty \frac{1}{(1+x)^n (\pi^2+\ln^2 x)} dx$

Integral $\int_0^{\pi} \frac{\cos(2018x)}{5-4\cos{x}}dx$

Integral $\int_0^\frac{\pi}{2} \arcsin(\sqrt{\sin x}) dx$

Integral $\int^{\frac{\pi}{6}}_{0}\ln^2(2\sin \theta)d\theta$

Integration trick $\int^{2\pi}_{0} f(a+ r \cos \theta, b+r\sin \theta)d\theta=2\pi f(a,b)$

Integral $\int_0^{\frac{\pi}{2}} x^2 \sqrt{\sin x} \, dx$

Integral $\int_0^1 \frac{(x^2+1)\ln(1+x)}{x^4-x^2+1}dx$

Integral $\int_0^1 \frac{\ln(1+x)}{1+x^3}dx$

Integral $\int_0^1 \frac{\arctan x}{x^2-x-1}dx$

Bounds for $\int_0^1 x^{\cos x+\sin x} dx$

Integral $\int_1^3 \frac{\log x}{x^2+3}dx$

Closed form for $\int_0^1 \frac {\log^n(x)}{(1-x)^m} dx$

Integral $\int_a^\infty \frac{\arctan(x+b)}{x^2+c}dx$