query: How to prove that $\int_0^1 x^{-x}\text{d}x=\sum_{n=1}^{\infty} n^{-n}$
positive_passages.docid: 24508
positive_passages.text: Series as an integral (sophomore's dream)
negative_passages.docid: 52902
negative_passages.text: How to prove $\int_{\mathbb R^d}{(1+|x|^{2})^{-m}dx}$ is integrable?
negative_passages.docid: 109901
negative_passages.text: Computing $\int_0^\infty\frac1{(x+1)(x+2)\cdots(x+n)}\mathrm dx $
negative_passages.docid: 107976
negative_passages.text: Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$
negative_passages.docid: 42805
negative_passages.text: Show that $\int\limits_0^1 \left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{\left(x^{x}\right)^{⋰}}}}\ \mathrm{d}x=\frac{\pi^2}{12}$.
negative_passages.docid: 61133
negative_passages.text: Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
negative_passages.docid: 68368
negative_passages.text: Evaluate $\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \ \mathrm{d}x$
negative_passages.docid: 34907
negative_passages.text: Show that $\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$
negative_passages.docid: 66404
negative_passages.text: Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$
negative_passages.docid: 25967
negative_passages.text: Prove $\csc(x)=\sum_{k=-\infty}^{\infty}\frac{(-1)^k}{x+k\pi}$
negative_passages.docid: 150305
negative_passages.text: $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
negative_passages.docid: 74955
negative_passages.text: Simplify result of $\int_0^{\infty} \frac{1}{1+x^n}dx$
negative_passages.docid: 16776
negative_passages.text: How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$
negative_passages.docid: 27705
negative_passages.text: Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
negative_passages.docid: 51837
negative_passages.text: Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
negative_passages.docid: 149362
negative_passages.text: Prove that $ \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = \int_{- \infty}^\infty \int_{- \infty}^\infty (F(x,y)-F_X(x) F_Y(y)) \,dx\,dy,$
negative_passages.docid: 54266
negative_passages.text: Prove that $\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$