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 15 How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$ 13 Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$ 10 Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$ 7 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 $\int_0^1\left(\sum_{k=n}^\infty {x^k\over k}\right)^2\,dx = \int_0^1 2x^{n-1}\log\left(1+{1\over\sqrt{x}}\right)\,dx.$

### Reputation (1,545)

 +10 Prove $\int_{0}^\infty \frac{1}{\Gamma(x)}\, \mathrm{d}x = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, \mathrm{d}x$ +10 Positive integral solutions of $3^x+4^y=5^z$ -2 How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$ +5 Two curious “identities” on $x^x$,$e$,and $\pi$

### Questions (14)

 58 Two curious “identities” on $x^x$,$e$,and $\pi$ 5 Root of a special polynomial 4 How to bound the following sum 4 Sphere Covering Problem 3 Some questions on the formation of the BSD conjecture

### Tags (40)

 35 calculus × 6 20 sequences-and-series × 3 33 definite-integrals × 4 15 closed-form 31 integration × 5 14 gamma-function × 3 28 real-analysis × 5 9 limits × 2 25 inequality × 6 5 roots × 3

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