便利さんの積分・級数botの積分,級数一覧

積分,級数一覧

一覧としてGitHubの これ があるのですがわたしはいちいち開いてみるのがめんどくさいので、ここでまとめておこうと思います。

積分

$$\int_{0}^{\infty}\frac{\sin x}{x}dx=\frac{\pi}{2}$$

$$\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{e^x+e^y+e^z-1}dxdydz=\frac{13}{4}\zeta(3)$$

$$\int_{0}^{1}\frac{\arctan\sqrt{2+x^2}}{(1+x^2)\sqrt{2+x^2}}dx=\frac{5\pi^2}{96}$$

$$\int_{0}^{1}\frac{1}{x^2+1}dx=\frac{\pi}{4}$$

$$\int_{0}^{\infty}\frac{\log^4x}{1+x^2}dx=\frac{5\pi^5}{64}$$

$$\int_{0}^{\frac{\pi}4}\cos\log\tan xdx=\frac{\pi}{4\cosh\frac{\pi}2}$$

$$\int_{0}^{1}\frac{\tanh^{-1}\frac{x}{2+\sqrt5}}{x}dx=\frac{\pi^2}{24}-\frac{3}{4}\log^2\phi$$

$$\int_{0}^{\infty}\frac{x\log\left(1-\frac{1}{\cosh x}\right)}{\cosh x}dx=-2\beta(2)\log2$$

$$\int_{0}^{\frac{\pi}2}\log\left(2\sin\frac{x}2\right)\log\left(2\cos\frac{x}{2}\right)dx=-\frac{\pi^3}{48}$$

$$\int_{0}^{2}\frac{1}{\sqrt{1+x^3}}dx=\frac{\Gamma\left(\frac{1}{3}\right)}{2^{\frac{4}{3}}\sqrt3\pi}$$

$$\int_{0}^{\infty}\frac{x\sinh x}{(1+\cosh^2 x)^2}dx=\frac{\pi}{4}\left(\log(1+\sqrt2)-\frac{1}{\sqrt2}\right)$$

$$\int_{0}^{\infty}\frac{\sinh x}{x\cosh \frac{x}{a}}dx=\log\cot\frac{\pi(1-a)}{4}$$

$$\int_{0}^{1}\frac{x\tanh^{-1}x}{3+x^2}dx=\frac{\pi^2}{72}$$

$$\int_{0}^{\infty}\log\left(\frac{1+x^2+x^4+x^6}{1+x^6}\right)dx=\pi(\sqrt2-1)$$

$$\int_{0}^{\frac{\pi}{2}}\arctan(2\tan^2x)dx=\pi\arctan\frac{1}2$$

$$\int_{0}^{\frac{\pi}2}\arctan\frac{\sin x}{2}dx=\frac{\pi^2}{12}-\frac{3\log^2\phi}{2}$$

$$\int_{0}^{\infty}\frac{\sin x}{x}\int_{0}^{x}\frac{\cos y}{y}dydx=0$$

$$\int_{0}^{1}\frac{\tanh^{-1}\sqrt{1-x^2}}{1-x}dx=\frac{3\pi^2}{8}$$

$$\int_{0}^{\frac{\pi}2}\log{\left(\frac{9}{16}+\cos^2x\right)}dx=0$$

$$\int_{0}^{\pi}\arctan\frac{\log\sin x}{x}dx=-\pi\arctan\frac{2\log2}{\pi}$$

$$\int_{1}^{\sqrt2}\frac{\arctan\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{96}$$

$$\int_{0}^{\infty}\frac{e^{-kx}\sin x}{1+x^2}dx=\int_{0}^{\infty}\frac{\sin x}{1+(x+k)^2}dx$$

$$\int_{0}^{1}\log\Gamma(x)dx=\frac{1}{2}\log2\pi$$

$$\int_{0}^{\frac{1}{2}}\frac{1}{x}\log(1+x)\log\left(\frac{1}{x}-1\right)dx=\frac{13}{24}\zeta(3)$$

$$\int_{0}^{\pi}\log\left(\frac{5}{4}+\cos x\right)dx=0$$

$$\int_{0}^{\frac{\pi}{2}}\frac{1+\cos 2x}{3+\cos 4x}dx=\frac{\pi}{4\sqrt2}$$

$$\int_{-\infty}^{\infty}\frac{\coth^{-1}\sqrt{x^2+\pi^2}}{\sqrt{x^2+\pi^2}}dx=\pi\arcsin\frac{1}{\pi}$$

$$\int_{0}^{1}\frac{x}{\tanh^{-1}x}dx=\frac{7\zeta(3)}{\pi^2}$$

$$\int_{0}^{\infty}x\log\tanh xdx=-\frac{7}{16}\zeta(3)$$

$$\int_{0}^{\frac{\pi}2}\frac{1}{1+\cos^4x}dx=\frac{\sqrt{1+\sqrt{2}}}{4}\pi$$

$$\mathrm{Re}\int_{0}^{\infty}H_{\frac{i}{x}}dx=\frac{\pi^3}{12}$$

$$\int_{0}^{\frac{\pi}{2}}\frac{\log\cos x}{x^2+\log^2\cos x}dx=\frac{\pi}{2}\left(1-\frac{1}{\log2}\right)$$

級数

$$\sum_{n=0}^{\infty}\frac{{2n\choose n}^3}{2^{6n}}=\frac{\pi}{\Gamma\left(\frac{3}{4}\right)^4}$$

$$\sum_{m,n\in\mathbb{Z}}q^{n^2+m^2}=1+4\sum_{n=0}^{\infty}\frac{(-1)^nq^{2n+1}}{1-q^{2n+1}}$$

$$\left(\sum_{n\in\mathbb{Z}}q^{n^2}\right)^4=1+8\sum_{\begin{array}{}{0< n}\\n \not\equiv0\ \mathrm{mod}4\end{array}}\frac{nq^n}{1-q^n}$$

$$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}=\frac{\pi}{4}$$

$$\sum_{0\leq n_1< n_2\leq n_3}\frac{1}{(n_1+\frac{1}{2})n_2(n_3+\frac{1}{2})}\frac{{2n_3\choose n_3}}{2^{2n_3}}=\frac{\pi^3}{3}$$

$$\sum_{0< n,m}\frac{1}{2^nnm^2}\left(\frac{n!m!}{(n+m)!}\right)^2=\frac{\pi^2\log2}{6}-\frac{13}{16}\zeta(3)$$

$$\sum_{0< n,m}\frac{1}{2^mnm^2}\left(\frac{n!m!}{(n+m)!}\right)^2=\frac{\log^32}{12}-\frac{\pi^2\log2}{8}+\frac{27}{32}\zeta(3)$$

$$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}6$$

$$\prod_{n\in\mathbb{Z}}\left(1+\frac{1}{\cosh2\pi n}\right)=2^{\frac{3}8}\sqrt{1+\sqrt2}$$

$$\sum_{n\in\mathbb{Z}}\frac{1}{\cosh^2\pi n}=\frac{1}{\pi}+\frac{\Gamma\left(\frac{1}{4}\right)^4}{8\pi^3}$$

$$\sum_{n\in\mathbb{Z}}\frac{1}{\cosh^2(\sqrt3\pi n)}=\frac{1}{\sqrt3\pi}+\frac{\Gamma\left(\frac{1}{3}\right)^6}{2^{\frac{11}{3}}\pi^4}(1+\sqrt3)$$

$$\sum_{n\in\mathbb{Z}}\frac{1}{(n+x)^2}=\frac{\pi^2}{\sin^2\pi x}$$

$$\sum_{n=1}^{\infty}\frac{1}{n(e^{2\pi n}-1)}= \frac{\log2}{2}+\log\Gamma\left(\frac{3}{4}\right) -\frac{\log\pi}{4}-\frac{\pi}{12} $$

$$\sum_{n=0}^{\infty}\frac{((2n)!)^3}{(n!)^6}\frac{6n+1}{2^{8n}}=\frac{4}{\pi}$$

$$\sum_{n=0}^{\infty}\frac{((2n)!)^3}{(n!)^6}\frac{42n+1}{2^{12n}}=\frac{16}{\pi}$$

$$\sum_{0< m< n}\frac{1}{m}\frac{(1-x)_m n!}{m!(1-x)_n)}\frac{1}{n^2}\frac{n!a!}{(a+n)!}=\frac{(1-x)_a}{a!}\sum_{a< n}\frac{1}{n^3(1-x)_n}$$

$$\sum_{n=1}^{\infty}(-1)^n\log\left(\frac{1}{n}+1\right)=\log\frac{2}{\pi}$$

$$\sum_{n=0}^{\infty}{2n\choose n}\frac{F_n}{2^{3n}}=\sqrt{\frac{2}5}$$

$$\sum_{n=0}^{\infty}\frac{2^{2n}}{(2n+1)^2{2n\choose n}}=2\beta(2)$$

$$\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)^3}=\frac{\pi^3}{32}$$

$$\sum_{n=1}^{\infty}\frac{H^2_n}{\varphi^{2n-1}}=\frac{\pi^2}{15}$$

$$\sum_{k=0}^{n}\frac{k}{n^k(n-k)!}=\frac{1}{(n-1)!}$$

$$\sum_{n=0}^{\infty}\mathrm{Li}_3\left(-e^{-(2n-1)\pi}\right)=-\frac{\pi^3}{720}$$

$$\sum_{n=0}^{\infty}{2n\choose n}^2\frac{1}{(2n+1)2^{4n}}=\frac{4\beta(2)}{\pi}$$

$$\sum_{n=0}^{\infty}\frac{2^{2n}}{(2n+1)^2{2n\choose n}}=2\beta(2)$$

$$\sum_{n\in\mathbb{Z}}e^{-\pi n^2}=\frac{\sqrt[4]{\pi}}{\Gamma\left(\frac{4}{3}\right)}$$

$$\sum_{n=1}^{\infty}\frac{H_n}{n^2}=2\zeta(3)$$

$$\sum_{n=1}^{\infty}\frac{1}{n^2{2n\choose n}}=\frac{\pi^2}{18}$$

$$\sum_{0< n< m}\frac{1}{2^nnm^2}\frac{{2n\choose n}}{{2m\choose m}}=\frac{\pi^2\log2}{18}-\frac{13}{48}\zeta(3)$$