query: Show that $\int_0^1 \prod_{n\geq 1} (1-x^n) \, dx = \frac{4\pi\sqrt{3}\sinh(\frac{\pi}{3}\sqrt{23})}{\sqrt{23}\cosh(\frac{\pi}{2}\sqrt{23})}$
positive_passages.docid: 88708
positive_passages.text: An integration of product $(1-x^n)$
negative_passages.docid: 99753
negative_passages.text: How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$?
negative_passages.docid: 281
negative_passages.text: How one can prove $\prod_{k=1}^{n-1} (\cos\theta-\cos(k\pi/n)) = 2^{1-n}\sin(n\theta)\csc\theta$?
negative_passages.docid: 100869
negative_passages.text: How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$
negative_passages.docid: 101371
negative_passages.text: show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$
negative_passages.docid: 54055
negative_passages.text: Proof $\prod_{i = 1}^n \frac{n + i}{2i-3} = 2^n(1-2n)$ using inducction
negative_passages.docid: 31792
negative_passages.text: show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $
negative_passages.docid: 12781
negative_passages.text: Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
negative_passages.docid: 47923
negative_passages.text: For $n\in\mathbb{N}$, evaluate $\int_0^n \cos(2\pi \lfloor x\rfloor\{x\})dx$
negative_passages.docid: 80553
negative_passages.text: Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$
negative_passages.docid: 117046
negative_passages.text: Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$
negative_passages.docid: 63510
negative_passages.text: Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 126236
negative_passages.text: General closed form of $\int_{0}^{\theta_0} \frac{1}{\sqrt{\cos \theta-\cos \theta_0}} d\theta$
negative_passages.docid: 36366
negative_passages.text: Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
negative_passages.docid: 109785
negative_passages.text: show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$