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Infiniticism
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50 votes
1 answer
3k views

Evaluate $\int_{-\infty}^\infty\frac{x}{\sin^2(\sqrt{x})\sinh^2\left(2\sqrt{2x}\right)+\pi^2\cos^2(\sqrt{x})\cosh^2\left(2\sqrt{2x}\right)}\mathrm dx$

34 votes
1 answer
2k views

Prove $\int_0^{\infty }\frac1{\sqrt{x}}\left(\frac{\cos(\pi x^2)}{\sinh (\pi x)}-\frac1{\pi x}\right)dx=\frac{1}{\sqrt{2}}\zeta(\frac{1}{2})$

21 votes
3 answers
2k views

Proving $\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx=\frac{3 \pi ^2}{160}$

17 votes
3 answers
623 views

Evaluate $\int_{(-\infty,\infty)^n}\frac{\prod_{k=1}^n \sin(a_k x_k)}{\prod_{k=1}^n x_k}\frac{\sin(\sum_{k=1}^n a_k x_k)}{\sum_{k=1}^n a_k x_k}$

15 votes
1 answer
711 views

A suprising conjectural closed-form of $\sum _{n=1}^{\infty } \frac{1}{n^4 2^n \binom{3 n}{n}}$ and integral variations

15 votes
1 answer
691 views

Evaluate hypergeometric $_6F_5\left(\{\frac12\}_3,\{1\}_3;\{\frac32\}_5;1\right)$

15 votes
3 answers
986 views

Prove that $_4F_3\left(\frac13,\frac13,\frac23,\frac23;1,\frac43,\frac43;1\right)=\frac{\Gamma \left(\frac13\right)^6}{36 \pi ^2}$

14 votes
2 answers
514 views

Proving $\int_0^\infty\left(\frac{x^xe^{-x}}{\Gamma(x+1)}-\frac1{\sqrt{2\pi x}}\right)dx=-\frac13$

13 votes
1 answer
1k views

Proving $\int_{\sqrt{5/7}}^1 \frac{(\pi-3\arctan\sqrt{\frac{2x^2-1}{3x^2-2}})\arctan x}{\sqrt{2x^2-1}(3x^2-1)} dx = \frac{\pi^3}{672}$

13 votes
1 answer
466 views

Proving $\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }\log \left| 1+e^{i x}+e^{i y}+e^{i z}\right| dxdydz=28 \pi \zeta (3)$

13 votes
3 answers
621 views

Showing $\int_0^{\infty } \frac{1}{e^{2 x}+2 e^x \cos (x)+1} \, dx=\frac{\log (2)}{2}-\pi \sum _{n=0}^{\infty } \frac{1}{e^{\pi (2 n+1)}+1}$

10 votes
1 answer
672 views

Evaluate $\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx$

10 votes
3 answers
659 views

Evaluate $\sum _{n=1}^{\infty } \frac{1}{n^5 2^n \binom{3 n}{n}}$ in terms of elementary constants

10 votes
1 answer
380 views

Prove $\, _6F_5\left(\{\frac12\}_5,\frac{5}{4};\frac{1}{4},\{1\}_4;-1\right)=\frac{2}{\Gamma \left(\frac{3}{4}\right)^4}$ and another

9 votes
1 answer
406 views

Evaluate $\int_0^{\infty } \frac{\tan ^{-1}\left(\sqrt{a^2+x^2}\right)}{\left(x^2+1\right)\sqrt{a^2+x^2}} \, dx$

9 votes
1 answer
297 views

Closed form of hypergeometric $\, _4F_3\left(\frac{3}{8},\frac{5}{8},\frac{7}{8},\frac{9}{8};\frac{5}{6},\frac{7}{6},\frac{9}{6};z\right)$

8 votes
1 answer
388 views

Fourier Legendre expansion for $\frac{\text{Li}_2(x)}{x}$

8 votes
2 answers
419 views

Evaluate $\int_0^{\infty } \Bigl( 2qe^{-x}-\frac{\sinh (q x)}{\sinh \left(\frac{x}{2}\right)} \Bigr) \frac{dx}x$

7 votes
3 answers
353 views

Proving $\sum _{k=1}^n \frac{(-1)^{k-1} 16^k (k-1)! k! (k+n-1)!}{((2 k)!)^2 (n-k)!}=\frac{4}{n}\sum _{k=1}^n \frac{1}{2 k-1}$

7 votes
3 answers
259 views

Asymptotic expansion of $f(x)=\sum _{n=1}^{\infty } \frac{\sin \left(\sqrt{n}x\right)}{n}$ at the origin

7 votes
2 answers
367 views

Evaluate $\int_0^1 \frac{\log ^2(x+1) \log \left(x^2+1\right)}{x^2+1} dx$

7 votes
3 answers
514 views

Evaluate $\int_0^{\infty } \log \left(\frac{a^2}{x^2}+1\right) \log \left(\frac{b^2}{x^2}+1\right) \log \left(\frac{c^2}{x^2}+1\right) \, dx$

7 votes
1 answer
321 views

Evaluate $\int_0^{\pi/2} \frac{\cos ((1-a) x)}{\cos ^{a-1}(x) (\cosh (2 b)-\cos (2 x))} \, dx$

7 votes
1 answer
238 views

Evaluate $\int _0^{2 \pi }\int _0^{2 \pi }\log (3-\cos (x+y)-\cos (x)-\cos (y))dxdy$

7 votes
3 answers
381 views

Proving $\int_0^1 \sqrt{x \left(\sqrt{-3 x^2+2 x+1}-x+1\right)} \, dx=\frac{7 \pi }{12 \sqrt{6}}$

7 votes
1 answer
217 views

$\sum _{n=0}^{\infty} \frac{1}{(n+1) (n+2)} \left(\frac{1}{\lfloor n \phi \rfloor +2}+\frac{1}{\lfloor n \phi ^{-1} \rfloor +2}\right)$

6 votes
2 answers
314 views

A curious identity on $\sum _{n=0}^{\infty } \frac{(-1)^n \cos \left(2 \pi \sqrt{2 n+1}\right)}{\sqrt{2 n+1}}$

6 votes
3 answers
456 views

Prove that $\sum _{k=1}^{n-1} \binom{n-1}{k} k^{k-1} (n-k)^{n-k-1}=n^{n-1}-n^{n-2}$

6 votes
1 answer
224 views

Prove that $\sum _{k=0}^n \left(\sum _{j=0}^k \binom{n}{j}\right)^3=\left(\frac{n}{2}+1\right) 8^n-\frac{3}{4} n 2^n \binom{2 n}{n}$

6 votes
1 answer
290 views

Show that $\int_0^1 \frac{x \text{csch}(a x)}{\sqrt{\cosh (2 a)-\cosh (2 a x)}} \mathrm dx=\frac{\pi \sin ^{-1}(\tanh (a))}{2 \sqrt{2} a^2 \sinh (a)}$