Question

Closed form for $\int \frac{1}{x^7 -1} dx$?

score 286 · Answer 1 · 2013-10-08 10:46:17

286

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Closed form for $\int \frac{1}{x^7 -1} dx$?

Published on 08 Oct 2013 - 10:46

score 1.1k · Answer 2 · 2013-10-09 08:10:54

1.1k

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Find a good candidate for a closed-form solution of this recurrence relation: $P(n-1)+n^2$.

Published on 09 Oct 2013 - 8:10

score 588 · Answer 3 · 2013-10-09 10:37:26

588

Views

Is there a closed form expression for $1 + x + x^{4} + x^{9}+x^{16}+x^{25} +..+x^{k^2}$?

Published on 09 Oct 2013 - 10:37

score 1.6k · Answer 4 · 2026-03-27 04:38:40

1.6k

Views

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Published on 27 Mar 2026 - 4:38

score 4.7k · Answer 5 · 2013-10-09 20:56:08

4.7k

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A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Published on 09 Oct 2013 - 20:56

score 553 · Answer 6 · 2026-03-29 07:29:19

553

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A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

Published on 29 Mar 2026 - 7:29

score 133 · Answer 7 · 2013-10-10 03:47:08

133

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Help calculating $\sum^{R}_{k=1} \bigl\lfloor{\sqrt { R^2-k^2}}\bigr\rfloor$

Published on 10 Oct 2013 - 3:47

score 373 · Answer 8 · 2026-03-30 03:38:27

373

Views

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

Published on 30 Mar 2026 - 3:38

score 106 · Answer 9 · 2026-04-01 13:31:16

106

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Is it possible to get a 'closed form' for $\sum_{k=0}^{n} a_k b_{n-k}$?

Published on 01 Apr 2026 - 13:31

score 3.6k · Answer 10 · 2026-03-29 10:49:03

3.6k

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Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$

Published on 29 Mar 2026 - 10:49

score 730 · Answer 11 · 2026-03-27 03:44:10

730

Views

Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$

Published on 27 Mar 2026 - 3:44

score 2.3k · Answer 12 · 2013-10-11 20:13:07

2.3k

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Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

Published on 11 Oct 2013 - 20:13

score 470 · Answer 13 · 2026-03-30 03:38:03

470

Views

Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$

Published on 30 Mar 2026 - 3:38

score 763 · Answer 14 · 2013-10-12 16:54:16

763

Views

Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$?

Published on 12 Oct 2013 - 16:54

score 468 · Answer 15 · 2013-10-12 17:25:21

468

Views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Published on 12 Oct 2013 - 17:25

List Question

Closed form for $\int \frac{1}{x^7 -1} dx$?

Find a good candidate for a closed-form solution of this recurrence relation: $P(n-1)+n^2$.

Is there a closed form expression for $1 + x + x^{4} + x^{9}+x^{16}+x^{25} +..+x^{k^2}$?

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

Help calculating $\sum^{R}_{k=1} \bigl\lfloor{\sqrt { R^2-k^2}}\bigr\rfloor$

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

Is it possible to get a 'closed form' for $\sum_{k=0}^{n} a_k b_{n-k}$?

Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$

Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$

Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$

Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$?

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

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