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### Questions (258)

 75 Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing 74 Evaluating $\lim_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ 44 Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ 42 Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$ 37 Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$

### Reputation (16,652)

 +5 Find the closed form of the digamma related series -2 Find the closed form of $\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$ +5 Prove that $\sum_{k=1}^{\infty} \large\frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$ +5 Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$

 47 The last digit of $2^{2006}$ 24 Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$ 21 Prove $\int_0^1 \frac{t^2-1}{(t^2+1)\log t}dt = 2\log\left( \frac{2\Gamma \left( \frac{5}{4}\right)}{\Gamma\left( \frac{3}{4}\right)}\right)$ 19 How to integrate $\int_1^e \ln{x} \, dx$ 18 Integral Contest

### Tags (79)

 342 calculus × 290 134 improper-integrals × 40 297 integration × 146 129 real-analysis × 262 195 definite-integrals × 89 52 inequality × 46 175 limits × 164 49 closed-form × 11 150 sequences-and-series × 148 48 elementary-number-theory × 3

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