Chris's sis the artist
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### Questions (295)

 90 Evaluating $\lim_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ 77 Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing 48 Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ 42 Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$ 41 Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$

### Reputation (18,964)

 +10 The last digit of $2^{2006}$ +5 Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$ +10 How to integrate $\int_1^e \ln{x} \, dx$ +5 Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

### Answers (240)

 52 The last digit of $2^{2006}$ 26 Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$ 22 Prove $\int_0^1 \frac{t^2-1}{(t^2+1)\log t}dt = 2\log\left( \frac{2\Gamma \left( \frac{5}{4}\right)}{\Gamma\left( \frac{3}{4}\right)}\right)$ 20 Evaluate $\int_0^{\pi/2}\frac{x^2\log^2{(\sin{x})}}{\sin^2x}dx$ 20 How to integrate $\int_1^e \ln{x} \, dx$

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 391 calculus × 330 151 improper-integrals × 43 325 integration × 178 139 real-analysis × 299 216 definite-integrals × 119 59 inequality × 47 209 limits × 167 58 closed-form × 12 164 sequences-and-series × 157 53 elementary-number-theory × 3

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