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 19 How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$ 14 Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$ 11 Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$ 8 Prove $\sum_{i=1}^{n}\frac{x_{i}^3}{x_{i+1}^2}\geq \sum_{i=1}^{n}\frac{x_{i}^2}{x_{i+1}}$ 7 How to prove that $\int_0^1\left(\sum\limits_{k=n}^\infty {x^k\over k}\right)^2\,dx = \int_0^1 2x^{n-1}\log\left(1+{1\over\sqrt{x}}\right)\,dx$

### Reputation (1,682)

 +10 How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$ +10 Computing $\int_0^{\infty} \frac{ \sqrt [3] {x+1}-\sqrt [3] {x}}{\sqrt{x}} \mathrm dx$ +15 Two curious “identities” on $x^x$,$e$,and $\pi$ +5 Some questions on the formation of the BSD conjecture

### Questions (15)

 67 Two curious “identities” on $x^x$,$e$,and $\pi$ 6 Square-free value of $n^2+1$ without large prime factors 5 Sphere Covering Problem 5 Root of a special polynomial 4 Some questions on the formation of the BSD conjecture

### Tags (40)

 42 calculus × 6 21 sequences-and-series × 3 40 definite-integrals × 4 19 closed-form 38 integration × 5 16 gamma-function × 3 30 real-analysis × 5 10 limits × 2 27 inequality × 6 6 roots × 3

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