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。
1. [int_0^1 {frac{{ln left( {1 + {x^{2 + sqrt 3 }}}
ight)}}{{1 + x}}dx} = frac{{{pi ^2}}}{{12}}left( {1 - sqrt 3 }
ight) + ln 2 cdot ln left( {1 + sqrt 3 }
ight)]
2. [int_0^1 {frac{{ln left( {1 + {x^{4 + sqrt {15} }}}
ight)}}{{1 + x}}dx} = frac{{{pi ^2}}}{{12}}left( {2 - sqrt {15} }
ight) + ln frac{{1 + sqrt 5 }}{2} cdot ln left( {2 + sqrt 3 }
ight) + + ln 2 cdot ln left( {sqrt 3 + sqrt 5 }
ight)]
3. [int_0^1 {frac{{ln left( {1 + {x^{6 + sqrt {35} }}}
ight)}}{{1 + x}}dx} = frac{{{pi ^2}}}{{12}}left( {3 - sqrt {35} }
ight) + ln frac{{1 + sqrt 5 }}{2} cdot ln left( {8 + 3sqrt 7 }
ight) + + ln 2 cdot ln left( {sqrt 5 + sqrt 7 }
ight)]
4. [int_0^1 {frac{{ar anh xln x}}{{xleft( {1 - x}
ight)left( {1 + x}
ight)}}dx} ]
5. [int_0^1 {frac{{arctan {x^{3 + sqrt 8 }}}}{{1 + {x^2}}}dx} ]
6. [int_0^{frac{pi }{3}} {x{{ln }^2}left( {2sin frac{x}{2}}
ight)dx} ]
7. [int_0^{frac{pi }{2}} {{{ln }^n}sin xdx} ]
8. [int_0^infty {frac{{sin x}}{{cosh x - cos x}} cdot frac{{{x^n}}}{{n!}}dx} ]
9. [int_0^infty {frac{{sin x}}{{cosh x + cos x}} cdot frac{{{x^n}}}{{n!}}dx} ]
10. [int_{frac{pi }{4}}^{frac{pi }{2}} {ln ln left( { an x}
ight)dx} ]
11. [int_0^infty {frac{{x - sin x}}{{left( {{pi ^2} + {x^2}}
ight){x^3}}}dx} = frac{1}{{2{pi ^3}}}left( {1 + frac{{{pi ^2}}}{2} - pi - frac{1}{{{e^pi }}}}
ight)]
12. [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. [prodlimits_{k = 4}^infty {left[ {1 - {{left( {frac{3}{k}}
ight)}^3}}
ight]} = frac{8}{{15561}} cdot frac{{cos left( {frac{{3sqrt 3 }}{2}pi }
ight)}}{pi }]
14. [sumlimits_{m = 1}^infty {sumlimits_{n = {2^{m - 1}}}^{{2^m} - 1} {frac{m}{{left( {2n + 1}
ight)left( {2n + 2}
ight)}}} } = 1 - gamma ]
15. [int_0^1 {frac{{left( {1 - x}
ight)ln x cdot {e^{ - x}}}}{{pi - x}}dx} ]
16. [mathop {lim }limits_{n o infty } left[ { - frac{1}{{2m}} + ln left( {frac{e}{m}}
ight) + sumlimits_{n = 2}^m {left( {frac{1}{n} - frac{{varsigma left( {1 - n}
ight)}}{{{m^n}}}}
ight)} }
ight] = gamma ]
17. [int_0^infty {frac{1}{{x{e^x}left( {{pi ^2} + {{ln }^2}x}
ight)}}dx} ]
18. [int_0^infty {frac{{ln left( {1 + {x^2}}
ight)}}{{{e^{2pi x}} - 1}}dx} ]
19. [sumlimits_{n = 1}^infty {frac{{sumlimits_{k = 1}^n {frac{1}{{{k^4}}}} }}{{{n^2}}}} ]
20. [int_0^1 {int_0^1 {int_0^1 {int_0^1 {frac{{dwdxdydz}}{{left( {wxy - 1}
ight)left( {zwx - 1}
ight)left( {yzw - 1}
ight)left( {xyz - 1}
ight)}}} } } } ]
21. [int_0^infty {frac{{{e^{ - {x^2}}}}}{{{pi ^2} + {{left( {gamma + x}
ight)}^2}}}dx} ]
22. [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{{3sqrt 3 }}{2}}
ight)]
23. [int_0^{2arcosh pi } {frac{{dx}}{{1 + frac{{{{sinh }^2}x}}{{{pi ^4}}}}}} ]
24. [int_0^infty {frac{{x{e^x}}}{{sqrt {4{e^x} - 3} left( {1 + 2{e^x} - sqrt {4{e^x} - 3} }
ight)}}dx} ]
25. [int_0^{frac{pi }{2}} {frac{{{{ln }^2}left( {2cos x}
ight)}}{{{{ln }^2}left( {2cos x}
ight) + {x^2}}}dx} ]
26. [mathop {lim }limits_{n o infty } frac{{n!}}{{{n^n}}}left( {sumlimits_{k = 0}^n {frac{{{n^k}}}{{k!}}} - sumlimits_{k = n + 1}^infty {frac{{{n^k}}}{{k!}}} }
ight)]
27. [sumlimits_{k = 1}^infty {frac{1}{{{k^2}}}cos left( {frac{9}{{kpi + sqrt {{k^2}{pi ^2} - 9} }}}
ight)} ]
28. [sumlimits_{n = - infty }^infty {frac{{{{left( { - 1}
ight)}^n}}}{{{{left( {npi + phi }
ight)}^2}}}cos left( {sqrt {{n^2}{pi ^2} + {a^2} - {phi ^2}} }
ight)} = frac{{acos acot phi + phi sin a}}{{asin phi }}]
29. [sumlimits_{n = 0}^infty {frac{{{n^2}{pi ^2} + {phi ^2}}}{{{{left( {{n^2}{pi ^2} - {phi ^2}}
ight)}^2}}}{{left( { - 1}
ight)}^n}cos left( {sqrt {{n^2}{pi ^2} + {a^2} - {phi ^2}} }
ight)} = frac{{cos sqrt {{a^2} - {phi ^2}} }}{{2{phi ^2}}} + frac{{acos acot phi + phi sin a}}{{2asin phi }}]
30. [sumlimits_{n = 0}^infty {frac{{{{left( { - 1}
ight)}^n}left( {n + frac{1}{2}}
ight)}}{{{{left( {n + frac{1}{3}}
ight)}^2}{{left( {n + frac{2}{3}}
ight)}^2}}}} cos left[ {pi sqrt {left( {n + frac{1}{6}}
ight)left( {n + frac{5}{6}}
ight)} }
ight] = {pi ^2}{e^{frac{{pi sqrt 3 }}{6}}}]
31. [int_{ - 1}^1 {frac{{arctan x}}{{1 + x}}ln left( {frac{{1 + {x^2}}}{2}}
ight)dx} ]
32. [int_0^1 {sin left( {pi x}
ight){x^x}{{left( {1 - x}
ight)}^{1 - x}}dx} ]
33. [int_0^{frac{pi }{2}} {{x^2}{{ln }^2}left( {2cos x}
ight)dx} ]
34. [int_0^{frac{pi }{2}} {frac{{{x^2}}}{{{x^2} + {{ln }^2}left( {2cos x}
ight)}}dx} ]
35. [int_0^{frac{pi }{2}} {frac{{{x^2}{{ln }^2}left( {2cos x}
ight)}}{{{x^2} + {{ln }^2}left( {2cos x}
ight)}}dx}]
36. [mathop {lim }limits_{n o infty } frac{n}{{ln n}}left[ {frac{1}{pi }{{left( {sumlimits_{k = 1}^n {sin frac{pi }{{sqrt {{n^2} + k} }}} }
ight)}^n} - frac{1}{{sqrt[4]{e}}}}
ight]]
37. [int_0^1 {{x^{ - x}}{{left( {1 - x}
ight)}^{ - 1 + x}}sin left( {pi x}
ight)dx = frac{pi }{e}} = 1.15573]
38. [int_0^{ + infty } {frac{{dx}}{{1 + xleft| {sin x}
ight|}}} ]
39. [int_0^{ + infty } {frac{{dx}}{{{x^2} + {{left( {{n^2}{x^2} - 1}
ight)}^2}}}}]
参阅：[1]http://tieba.baidu.com/p/2114477017
[2]http://tieba.baidu.com/p/3148596990
[3]http://tieba.baidu.com/p/2121721064