西西爆难积分

在这里，主要展示西西12年7月在百度贴吧数学吧中贴出的30个积分题的求解，本文中主要参考的是自己以前的摘录，不知何故，与西哥的版本有些出入，但基本包含了西哥的所有问题，来源于网友的解答均会注明出处。

\int_{0}^{1} \frac{\ln (1 + x^{2 + \sqrt{3}})}{1 + x} d x = \frac{π^{2}}{12} (1 - \sqrt{3}) + \ln 2 \cdot \ln (1 + \sqrt{3})

$\int_0^1 {\frac{{\ln \left( {1 + {x^{2 + \sqrt 3 }}} \right)}}{{1 + x}}dx} = \frac{{{\pi ^2}}}{{12}}\left( {1 - \sqrt 3 } \right) + \ln 2 \cdot \ln \left( {1 + \sqrt 3 } \right)$

\int_{0}^{1} \frac{\ln (1 + x^{4 + \sqrt{15}})}{1 + x} d x = \frac{π^{2}}{12} (2 - \sqrt{15}) + \ln \frac{1 + \sqrt{5}}{2} \cdot \ln (2 + \sqrt{3}) + + \ln 2 \cdot \ln (\sqrt{3} + \sqrt{5})

$\int_0^1 {\frac{{\ln \left( {1 + {x^{4 + \sqrt {15} }}} \right)}}{{1 + x}}dx} = \frac{{{\pi ^2}}}{{12}}\left( {2 - \sqrt {15} } \right) + \ln \frac{{1 + \sqrt 5 }}{2} \cdot \ln \left( {2 + \sqrt 3 } \right) + + \ln 2 \cdot \ln \left( {\sqrt 3 + \sqrt 5 } \right)$

\int_{0}^{1} \frac{\ln (1 + x^{6 + \sqrt{35}})}{1 + x} d x = \frac{π^{2}}{12} (3 - \sqrt{35}) + \ln \frac{1 + \sqrt{5}}{2} \cdot \ln (8 + 3 \sqrt{7}) + + \ln 2 \cdot \ln (\sqrt{5} + \sqrt{7})

$\int_0^1 {\frac{{\ln \left( {1 + {x^{6 + \sqrt {35} }}} \right)}}{{1 + x}}dx} = \frac{{{\pi ^2}}}{{12}}\left( {3 - \sqrt {35} } \right) + \ln \frac{{1 + \sqrt 5 }}{2} \cdot \ln \left( {8 + 3\sqrt 7 } \right) + + \ln 2 \cdot \ln \left( {\sqrt 5 + \sqrt 7 } \right)$

\int_{0}^{1} \frac{a r \tanh x \ln x}{x (1 - x) (1 + x)} d x

$\int_0^1 {\frac{{ar\tanh x\ln x}}{{x\left( {1 - x} \right)\left( {1 + x} \right)}}dx}$

\int_{0}^{1} \frac{\arctan x^{3 + \sqrt{8}}}{1 + x^{2}} d x

$\int_0^1 {\frac{{\arctan {x^{3 + \sqrt 8 }}}}{{1 + {x^2}}}dx}$

\int_{0}^{\frac{π}{3}} x \ln^{2} (2 \sin \frac{x}{2}) d x

$\int_0^{\frac{\pi }{3}} {x{{\ln }^2}\left( {2\sin \frac{x}{2}} \right)dx}$

\int_{0}^{\frac{π}{2}} \ln^{n} \sin x d x

$\int_0^{\frac{\pi }{2}} {{{\ln }^n}\sin xdx}$

\int_{0}^{\infty} \frac{\sin x}{\cosh x - \cos x} \cdot \frac{x^{n}}{n!} d x

$\int_0^\infty {\frac{{\sin x}}{{\cosh x - \cos x}} \cdot \frac{{{x^n}}}{{n!}}dx}$

\int_{0}^{\infty} \frac{\sin x}{\cosh x + \cos x} \cdot \frac{x^{n}}{n!} d x

$\int_0^\infty {\frac{{\sin x}}{{\cosh x + \cos x}} \cdot \frac{{{x^n}}}{{n!}}dx}$

10.

\int_{\frac{π}{4}}^{\frac{π}{2}} \ln \ln (\tan x) d x

$\int_{\frac{\pi }{4}}^{\frac{\pi }{2}} {\ln \ln \left( {\tan x} \right)dx}$

11.

\int_{0}^{\infty} \frac{x - \sin x}{(π^{2} + x^{2}) x^{3}} d x = \frac{1}{2 π^{3}} (1 + \frac{π^{2}}{2} - π - \frac{1}{e^{π}})

$\int_0^\infty {\frac{{x - \sin x}}{{\left( {{\pi ^2} + {x^2}} \right){x^3}}}dx} = \frac{1}{{2{\pi ^3}}}\left( {1 + \frac{{{\pi ^2}}}{2} - \pi - \frac{1}{{{e^\pi }}}} \right)$

12.

\int_{0}^{\infty} \frac{1}{a^{2} + x^{2}} \frac{x - \sin x}{x^{3}} d x = \frac{a^{2} - 2 a + 2 - 2 e^{- a}}{4 a^{3}}, a > 0

$\int_0^\infty {\frac{1}{{{a^2} + {x^2}}}\frac{{x - \sin x}}{{{x^3}}}dx} = \frac{{{a^2} - 2a + 2 - 2{e^{ - a}}}}{{4{a^3}}},a > 0$

13.

\prod_{k = 4}^{\infty} [1 - {(\frac{3}{k})}^{3}] = \frac{8}{15561} \cdot \frac{\cos (\frac{3 \sqrt{3}}{2} π)}{π}

$\prod\limits_{k = 4}^\infty {\left[ {1 - {{\left( {\frac{3}{k}} \right)}^3}} \right]} = \frac{8}{{15561}} \cdot \frac{{\cos \left( {\frac{{3\sqrt 3 }}{2}\pi } \right)}}{\pi }$

14.

\sum_{m = 1}^{\infty} \sum_{n = 2^{m - 1}}^{2^{m} - 1} \frac{m}{(2 n + 1) (2 n + 2)} = 1 - γ

$\sum\limits_{m = 1}^\infty {\sum\limits_{n = {2^{m - 1}}}^{{2^m} - 1} {\frac{m}{{\left( {2n + 1} \right)\left( {2n + 2} \right)}}} } = 1 - \gamma$

15.

\int_{0}^{1} \frac{(1 - x) \ln x \cdot e^{- x}}{π - x} d x

$\int_0^1 {\frac{{\left( {1 - x} \right)\ln x \cdot {e^{ - x}}}}{{\pi - x}}dx}$

16.

lim_{n \to \infty} [- \frac{1}{2 m} + \ln (\frac{e}{m}) + \sum_{n = 2}^{m} (\frac{1}{n} - \frac{ς (1 - n)}{m^{n}})] = γ

$\mathop {\lim }\limits_{n \to \infty } \left[ { - \frac{1}{{2m}} + \ln \left( {\frac{e}{m}} \right) + \sum\limits_{n = 2}^m {\left( {\frac{1}{n} - \frac{{\varsigma \left( {1 - n} \right)}}{{{m^n}}}} \right)} } \right] = \gamma$

17.

\int_{0}^{\infty} \frac{1}{x e^{x} (π^{2} + \ln^{2} x)} d x

$\int_0^\infty {\frac{1}{{x{e^x}\left( {{\pi ^2} + {{\ln }^2}x} \right)}}dx}$

18.

\int_{0}^{\infty} \frac{\ln (1 + x^{2})}{e^{2 π x} - 1} d x

$\int_0^\infty {\frac{{\ln \left( {1 + {x^2}} \right)}}{{{e^{2\pi x}} - 1}}dx}$

19.

\sum_{n = 1}^{\infty} \frac{\sum_{k = 1}^{n} \frac{1}{k^{4}}}{n^{2}}

$\sum\limits_{n = 1}^\infty {\frac{{\sum\limits_{k = 1}^n {\frac{1}{{{k^4}}}} }}{{{n^2}}}}$

20.

\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{d w d x d y d z}{(w x y - 1) (z w x - 1) (y z w - 1) (x y z - 1)}

$\int_0^1 {\int_0^1 {\int_0^1 {\int_0^1 {\frac{{dwdxdydz}}{{\left( {wxy - 1} \right)\left( {zwx - 1} \right)\left( {yzw - 1} \right)\left( {xyz - 1} \right)}}} } } }$

21.

\int_{0}^{\infty} \frac{e^{- x^{2}}}{π^{2} + {(γ + x)}^{2}} d x

$\int_0^\infty {\frac{{{e^{ - {x^2}}}}}{{{\pi ^2} + {{\left( {\gamma + x} \right)}^2}}}dx}$

22.

\sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \dots}}}}} = \frac{2}{\sqrt{3}} \cos (\frac{1}{3} \arccos \frac{3 \sqrt{3}}{2})

$\sqrt[3]{{1 + \sqrt[3]{{1 + \sqrt[3]{{1 + \sqrt[3]{{1 + \sqrt[3]{{1 + \cdots }}}}}}}}}} = \frac{2}{{\sqrt 3 }}\cos \left( {\frac{1}{3}\arccos \frac{{3\sqrt 3 }}{2}} \right)$

23.

\int_{0}^{2 a r \cosh π} \frac{d x}{1 + \frac{\sinh^{2} x}{π^{4}}}

$\int_0^{2ar\cosh \pi } {\frac{{dx}}{{1 + \frac{{{{\sinh }^2}x}}{{{\pi ^4}}}}}}$

24.

\int_{0}^{\infty} \frac{x e^{x}}{\sqrt{4 e^{x} - 3} (1 + 2 e^{x} - \sqrt{4 e^{x} - 3})} d x

$\int_0^\infty {\frac{{x{e^x}}}{{\sqrt {4{e^x} - 3} \left( {1 + 2{e^x} - \sqrt {4{e^x} - 3} } \right)}}dx}$

25.

\int_{0}^{\frac{π}{2}} \frac{\ln^{2} (2 \cos x)}{\ln^{2} (2 \cos x) + x^{2}} d x

$\int_0^{\frac{\pi }{2}} {\frac{{{{\ln }^2}\left( {2\cos x} \right)}}{{{{\ln }^2}\left( {2\cos x} \right) + {x^2}}}dx}$

26.

lim_{n \to \infty} \frac{n!}{n^{n}} (\sum_{k = 0}^{n} \frac{n^{k}}{k!} - \sum_{k = n + 1}^{\infty} \frac{n^{k}}{k!})

$\mathop {\lim }\limits_{n \to \infty } \frac{{n!}}{{{n^n}}}\left( {\sum\limits_{k = 0}^n {\frac{{{n^k}}}{{k!}}} - \sum\limits_{k = n + 1}^\infty {\frac{{{n^k}}}{{k!}}} } \right)$

27.

\sum_{k = 1}^{\infty} \frac{1}{k^{2}} \cos (\frac{9}{k π + \sqrt{k^{2} π^{2} - 9}})

$\sum\limits_{k = 1}^\infty {\frac{1}{{{k^2}}}\cos \left( {\frac{9}{{k\pi + \sqrt {{k^2}{\pi ^2} - 9} }}} \right)}$

28.

\sum_{n = - \infty}^{\infty} \frac{{(- 1)}^{n}}{{(n π + ϕ)}^{2}} \cos (\sqrt{n^{2} π^{2} + a^{2} - ϕ^{2}}) = \frac{a \cos a \cot ϕ + ϕ \sin a}{a \sin ϕ}

$\sum\limits_{n = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^n}}}{{{{\left( {n\pi + \phi } \right)}^2}}}\cos \left( {\sqrt {{n^2}{\pi ^2} + {a^2} - {\phi ^2}} } \right)} = \frac{{a\cos a\cot \phi + \phi \sin a}}{{a\sin \phi }}$

29.

\sum_{n = 0}^{\infty} \frac{n^{2} π^{2} + ϕ^{2}}{{(n^{2} π^{2} - ϕ^{2})}^{2}} {(- 1)}^{n} \cos (\sqrt{n^{2} π^{2} + a^{2} - ϕ^{2}}) = \frac{\cos \sqrt{a^{2} - ϕ^{2}}}{2 ϕ^{2}} + \frac{a \cos a \cot ϕ + ϕ \sin a}{2 a \sin ϕ}

$\sum\limits_{n = 0}^\infty {\frac{{{n^2}{\pi ^2} + {\phi ^2}}}{{{{\left( {{n^2}{\pi ^2} - {\phi ^2}} \right)}^2}}}{{\left( { - 1} \right)}^n}\cos \left( {\sqrt {{n^2}{\pi ^2} + {a^2} - {\phi ^2}} } \right)} = \frac{{\cos \sqrt {{a^2} - {\phi ^2}} }}{{2{\phi ^2}}} + \frac{{a\cos a\cot \phi + \phi \sin a}}{{2a\sin \phi }}$

30.

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} (n + \frac{1}{2})}{{(n + \frac{1}{3})}^{2} {(n + \frac{2}{3})}^{2}} \cos [π \sqrt{(n + \frac{1}{6}) (n + \frac{5}{6})}] = π^{2} e^{\frac{π \sqrt{3}}{6}}

$\sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^n}\left( {n + \frac{1}{2}} \right)}}{{{{\left( {n + \frac{1}{3}} \right)}^2}{{\left( {n + \frac{2}{3}} \right)}^2}}}} \cos \left[ {\pi \sqrt {\left( {n + \frac{1}{6}} \right)\left( {n + \frac{5}{6}} \right)} } \right] = {\pi ^2}{e^{\frac{{\pi \sqrt 3 }}{6}}}$

31.

\int_{- 1}^{1} \frac{\arctan x}{1 + x} \ln (\frac{1 + x^{2}}{2}) d x

$\int_{ - 1}^1 {\frac{{\arctan x}}{{1 + x}}\ln \left( {\frac{{1 + {x^2}}}{2}} \right)dx}$

32.

\int_{0}^{1} \sin (π x) x^{x} {(1 - x)}^{1 - x} d x

$\int_0^1 {\sin \left( {\pi x} \right){x^x}{{\left( {1 - x} \right)}^{1 - x}}dx}$

33.

\int_{0}^{\frac{π}{2}} x^{2} \ln^{2} (2 \cos x) d x

$\int_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {2\cos x} \right)dx}$

34.

\int_{0}^{\frac{π}{2}} \frac{x^{2}}{x^{2} + \ln^{2} (2 \cos x)} d x

$\int_0^{\frac{\pi }{2}} {\frac{{{x^2}}}{{{x^2} + {{\ln }^2}\left( {2\cos x} \right)}}dx}$

35.

\int_{0}^{\frac{π}{2}} \frac{x^{2} \ln^{2} (2 \cos x)}{x^{2} + \ln^{2} (2 \cos x)} d x

$\int_0^{\frac{\pi }{2}} {\frac{{{x^2}{{\ln }^2}\left( {2\cos x} \right)}}{{{x^2} + {{\ln }^2}\left( {2\cos x} \right)}}dx}$

36.

lim_{n \to \infty} \frac{n}{\ln n} [\frac{1}{π} {(\sum_{k = 1}^{n} \sin \frac{π}{\sqrt{n^{2} + k}})}^{n} - \frac{1}{\sqrt[4]{e}}]

$\mathop {\lim }\limits_{n \to \infty } \frac{n}{{\ln n}}\left[ {\frac{1}{\pi }{{\left( {\sum\limits_{k = 1}^n {\sin \frac{\pi }{{\sqrt {{n^2} + k} }}} } \right)}^n} - \frac{1}{{\sqrt[4]{e}}}} \right]$

37.