# Chris's wise sister

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 31 The last digit of $2^{2006}$ 13 How to integrate $\int_1^e \ln{x} \, dx$ 11 Prove $\int_0^1 \frac{t^2-1}{(t^2+1)\ln t}dt = 2\log\left( \frac{2\Gamma \left( \frac{5}{4}\right)}{\Gamma\left( \frac{3}{4}\right)}\right)$ 11 Inequality $a+b+c \geqslant abc +2$ 10 Prove the following identity

# 11,490 Reputation

 +5 Methods to evaluate $\int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx}$ +5 An interesting sum to infinity +5 Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$ +5 Evaluate $\int\sin(\sin x)~dx$

# 197 Questions

 60 Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing 30 Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$ 30 Prove that $\sum_{k=1}^{\infty} \large\frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$ 28 $\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ 21 Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$

# 63 Tags

 145 calculus × 211 78 real-analysis × 197 102 integral × 77 49 integration × 36 99 sequences-and-series × 110 43 definite-integral × 46 98 limit × 147 35 inequality × 42 93 homework × 36 34 trigonometry × 25

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