Question

Is there a nice general formula for $\int \frac{dx}{x^n-1}$ and/or $\int \frac{dx}{\Phi_n(x)}$?

score 946 · Answer 1 · 2026-05-10 17:18:10

946

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Is there a nice general formula for $\int \frac{dx}{x^n-1}$ and/or $\int \frac{dx}{\Phi_n(x)}$?

Published on 10 May 2026 - 17:18

score 195 · Answer 2 · 2026-05-10 16:12:36

195

Views

How to define orientation mapping for parametrization of a 1-manifold in a 1 dimensional space

Published on 10 May 2026 - 16:12

score 112 · Answer 3 · 2026-04-14 07:02:39

112

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Measures uniquely determined by their restriction to $ C_0^\infty $

Published on 14 Apr 2026 - 7:02

score 70 · Answer 4 · 2026-05-10 17:00:39

70

Views

Show $ \int g$ d$\mathbb{P}_n = \frac{1}{n} \sum_{i=1}^n g(y_i) $ for any measurable function $g$.

Published on 10 May 2026 - 17:00

score 91 · Answer 5 · 2026-04-14 23:23:46

91

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Comparison test on $ \int_2 ^\infty \frac{\ln(t)}{t^{3/2}} \ dt$

Published on 14 Apr 2026 - 23:23

score 46 · Answer 6 · 2026-04-25 10:39:46

$f:\Omega \rightarrow \mathbb{R}$ integrable $\Leftrightarrow$ $\exists g \ge 0$ so that $\int_{|f|\ge g}|f| \, \mathrm{d}\mu < \varepsilon$

score 79 · Answer 7 · 2026-04-15 23:30:03

79

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$ \int_2^{\infty}\frac{1}{x\sqrt{x^2-4}}dx$ convergence verification

Published on 15 Apr 2026 - 23:30

score 89 · Answer 8 · 2026-05-06 03:11:05

89

Views

$n^2 \int_0^1 (1-x)^n \sin(\pi x) \mathrm{d}x$

Published on 06 May 2026 - 3:11

score 56 · Answer 9 · 2026-04-23 06:43:27

56

Views

Let $f(x)=x$ for $x\in [0,1]$ Then show $\lim _{n\rightarrow \infty } L(f,P_n)=\lim _{n\rightarrow \infty } U(f,P_n)=\dfrac {1} {2}$

Published on 23 Apr 2026 - 6:43

score 172 · Answer 10 · 2026-04-15 13:37:46

172

Views

Integral $\int_0^1x^{\frac{1}{x}}{\rm d}x$

Published on 15 Apr 2026 - 13:37

score 177 · Answer 11 · 2026-04-26 00:27:13

177

Views

How to integrate $\cos \left(\frac{y-x}{y+x}\right)$?

Published on 26 Apr 2026 - 0:27

score 8.3k · Answer 12 · 2026-05-11 05:13:55

8.3k

Views

Finding the probability density function of $Z = X + Y$

Published on 11 May 2026 - 5:13

score 94 · Answer 13 · 2026-04-15 17:04:00

94

Views

Estimate a value of $\int_2^\infty \frac{x^k e^{-x}}{x \left( \log(x) \right)^2}\, dx $

Published on 15 Apr 2026 - 17:04

score 85 · Answer 14 · 2026-04-17 09:48:43

85

Views

Equivalent integral results for $\int \sin^3(x) \rm dx$

Published on 17 Apr 2026 - 9:48

score 409 · Answer 15 · 2026-05-11 08:01:55

409

Views

$ y = \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^4} dx $

Published on 11 May 2026 - 8:01

List Question

Is there a nice general formula for $\int \frac{dx}{x^n-1}$ and/or $\int \frac{dx}{\Phi_n(x)}$?

How to define orientation mapping for parametrization of a 1-manifold in a 1 dimensional space

Measures uniquely determined by their restriction to $ C_0^\infty $

Show $ \int g$ d$\mathbb{P}_n = \frac{1}{n} \sum_{i=1}^n g(y_i) $ for any measurable function $g$.

Comparison test on $ \int_2 ^\infty \frac{\ln(t)}{t^{3/2}} \ dt$

$f:\Omega \rightarrow \mathbb{R}$ integrable $\Leftrightarrow$ $\exists g \ge 0$ so that $\int_{|f|\ge g}|f| \, \mathrm{d}\mu < \varepsilon$

$ \int_2^{\infty}\frac{1}{x\sqrt{x^2-4}}dx$ convergence verification

$n^2 \int_0^1 (1-x)^n \sin(\pi x) \mathrm{d}x$

Let $f(x)=x$ for $x\in [0,1]$ Then show $\lim _{n\rightarrow \infty } L(f,P_n)=\lim _{n\rightarrow \infty } U(f,P_n)=\dfrac {1} {2}$

Integral $\int_0^1x^{\frac{1}{x}}{\rm d}x$

How to integrate $\cos \left(\frac{y-x}{y+x}\right)$?

Finding the probability density function of $Z = X + Y$

Estimate a value of $\int_2^\infty \frac{x^k e^{-x}}{x \left( \log(x) \right)^2}\, dx $

Equivalent integral results for $\int \sin^3(x) \rm dx$

$ y = \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^4} dx $

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