204 A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$ 175 Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$ 156 Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ 119 Prove that $\frac{100!}{50!\cdot2^{50}} \in \Bbb{Z}$ 105 How can a “proper” function have a vertical slope?

### Reputation (130,056)

 +10 Prove that $\displaystyle \lim_{n\to\infty}\underline{\int_a^b}f_n(x)\ \text{d}x=0$ +10 Prove that $\int_0^{\infty} \frac{1-\cos(at)}{t^{1+\alpha}} dt = \frac{\pi}{2 \Gamma(\alpha+1) \sin (\alpha \pi /2 )} |a|^{\alpha}$ +8 Proving $\sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i-x_j|}\le \sum_{i=1}^n\sum_{j=1}^n\sqrt{|x_i+x_j|}$. +10 How much does symbolic integration mean to mathematics?

### Questions (39)

 188 Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$ 68 How much does symbolic integration mean to mathematics? 54 Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$ 50 Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi}$ 47 Polynomial equations $p(A, B) = 0$ for matrices that ensure $AB = BA$

### Tags (447)

 4k calculus × 386 1k limits × 200 4k integration × 458 931 improper-integrals × 107 2k definite-integrals × 235 848 closed-form × 51 2k real-analysis × 473 508 complex-analysis × 117 2k sequences-and-series × 331 444 probability × 156

### Bookmarks (552)

 1412 Visually stunning math concepts which are easy to explain 486 Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$ 426 Why does $1+2+3+\cdots = -\frac{1}{12}$? 326 How can you prove that a function has no closed form integral? 300 Limit of sequence of growing matrices