AOPS论坛上100+100个积分

100+10 rare and irresistible integrals

I bring you many beautiful integrals that I have collected over time, I hope you enjoy them as much as I do.
If you want to answer one of these integrals, please hide your answer.
#passion for this #Enjoy :showoff: :-D :weightlift: :stretcher:

1. Coxeter Integrals $\int_0^{\frac{\pi }{2}} {\arccos \left( {\frac{{\cos \theta }}{{1 + 2\cos \theta }}} \right)d\theta = \frac{5}{{24}}\pi ^2 }$

2. $\int_0^{\frac{\pi }{2}} {\arccos \left( {\frac{1}{{1 + 2\cos \theta }}} \right)d\theta = \frac{1}{8}\pi ^2 }$

3. $\int_0^{\frac{\pi }{2}} {\arccos \left( {\frac{{1 - \cos \theta }}{{2\cos \theta }}} \right)d\theta = \frac{{11}}{{72}}\pi ^2 }$

4. For any $n$ natural number. Show that $\int\limits_{0}^{2\pi }{\frac{\left( 1+2\cos x \right)^{n}\cos nx}{3+2\cos x}dx}=\frac{2\pi }{\sqrt{5}}\left( 3-\sqrt{5} \right)^{n}$

5. Let $0<a<1$ Prove that $\int\limits_{0}^{2\pi }{\frac{\cos ^{2}3x}{1+a^{2}-2a\cos 2x}dx}=\frac{a^{2}-a+1}{1-a}\pi$

6. For $a>1$ Prove that $\int\limits_{-\pi }^{\pi }{\frac{x\sin x}{1+a^{2}-2a\cos x}dx}=\frac{\pi }{2}\ln \left( 1+\frac{1}{a} \right)$

7. $\int\limits_{0}^{1}{\frac{\ln \ln \frac{1}{x}}{\left( 1+x \right)^{2}}dx}=\frac{1}{2}\left( \ln \pi -\ln 2-\gamma \right)$

8. $\int_{0}^{+\infty }{\frac{\sinh x}{\cosh ^{2}x}\frac{dx}{x}}=\frac{4G}{\pi }$ where $G$ is the Catalan's constant

9. Let $z$ be a real number. Show that $\displaystyle\frac{1}{2\pi}\int_0^{2\pi}\log|z-e^{i\theta}|\,d\theta = \left\{ \begin{array}{ll} 0 & \text{ si }|z|<1\\ \log|z| & \text{ si }|z|\ge1 \end{array} \right.$

10. $\int\limits_{0}^{+\infty }{\exp \left( -a^{2}x\left( \frac{x-6}{x-2} \right)^{2} \right)\frac{dx}{\sqrt{x}}}=\frac{\sqrt{\pi }}{a}$

11. Let $\alpha >0$ Prove that $I\left( \alpha \right)=\int\limits_{0}^{\frac{\pi }{2}}{\arctan \left( \frac{2\alpha \sin ^{2}x}{\alpha ^{2}-1+\cos ^{2}x} \right)dx}=\pi \arctan \left( \frac{1}{2\alpha } \right)$

12. $\int\limits_0^1 {\frac{{\log \left( {1 - x} \right)}}{x} \cdot \frac{{2z}}{{\log ^2 x + \left( {2\pi z} \right)^2 }}dx} = - \log \left( {\frac{{z!e^z }}{{z^z \sqrt {2\pi z} }}} \right),\;\;\operatorname{Re} \left( z \right) > 0$

13.$\int\limits_{0}^{1}{\frac{1-x}{\log x}\cdot \left( x+x^{2}+x^{2^{2}}+... \right)dx}$

14. Let $a_k > 0$ and $a_0 > \sum\limits_{k = 1}^n {a_k }$. Show that $\int\limits_0^{ + \infty } {\prod\limits_{k = 0}^n {\frac{{\sin \left( {a_k x} \right)}}{x}dx} } = \frac{\pi }{2}\prod\limits_{k = 1}^n {a_k }$

15. Let $0 < z < 1,\alpha > 0,\beta \in {\Bbb C}$
$\int\limits_0^{ + \infty } {\sin \left( {\alpha t^{\frac{1}{z}} + \beta } \right)} dt = \frac{{\Gamma \left( {z + 1} \right)}}
{{\alpha ^z }}\sin \left( {\frac{{\pi z}}{2} + \beta } \right)$

16. $\operatorname{Re}\left( \alpha \right)\ge 1$
$\int\limits_{ - \infty }^{ + \infty } {\left| {\sin x} \right|^{\alpha - 1} \frac{{\sin x}}{x}} dx = 2^{\alpha - 1} \frac{{\Gamma ^2 \left( {\frac{\alpha }{2}} \right)}}{{\Gamma \left( \alpha \right)}}$

17. $\int\limits_0^1 {\sin \left( {\pi x} \right)} x^x \left( {1 - x} \right)^{1 - x} dx = \frac{{\pi e}}{{24}}$

18. $\int\limits_0^{\frac{\pi }{4}} {\frac{{x^3 }}{{\sin ^2 x}}} dx = \frac{{3\pi }}{4}G - \frac{{\pi ^3 }}{{64}} + \frac{{3\pi ^2 }}
{{32}}\log 2 - \frac{{105}}{{64}}\varsigma \left( 3 \right)$

19. Let $\theta > 0$
$\int\limits_{ - \infty }^{ + \infty } {\frac{{\left| {\cos \theta x} \right|}}{{1 + x^2 }}dx} = 4\cosh \theta \arctan e^{ - \theta }$

20. Let $\alpha \geqslant 0,\theta \in {\Bbb C}\backslash \pi {\Bbb Z}$
$\int\limits_{ - \infty }^{ + \infty } {\frac{{\cos \alpha x}}{{1 + 2\cos \theta x + x^2 }}dx} = \frac{\pi }{{\sin \theta }}\frac{{\cos \left( {\alpha \cos \theta } \right)}}{{e^{\alpha \sin \theta } }}$

21. Show that $\int_{0}^{\frac{\pi }{2}}{\frac{d\theta }{1+\sin ^{2}\tan \theta }}=\frac{\pi }{2\sqrt{2}}\left( \frac{e^{2}+3-2\sqrt{2}}{e^{2}-3+2\sqrt{2}} \right)$

22. Let $\theta \in \left[ 0,\frac{\pi }{2} \right)$ Prove that $\int\limits_{-\infty }^{\infty }{\frac{\arctan x}{x^{2}-2x\sin \theta +1}dx}$

23. Given the function $y\left(x\right):\left [0,1\right]\to\left [0,1\right]$ continuous and decreasing such that $x^{a}-x^{b} = y^{a}-y^{b}$. Compute $\int\limits_{0}^{1}{\frac{\ln \left( y\left( x \right) \right)}{x}dx}$

24. $\int_{0}^{1}\left ( -1 \right )^{\left [ 1994x \right ] + \left [ 1995x \right ]}\binom{1993}{\left [ 1994x \right ]}\binom{1994}{\left [ 1995x \right ]}dx$

25. $\int\limits_0^1 {\frac{{dx}}{{1 + {}_2F_1 \left( {\frac{1}{n},x;\frac{1}{n};\frac{1}{n}} \right)}}} = \frac{{\log \left( {\frac{{2n}}{{2n - 1}}} \right)}}{{\log \left( {\frac{n}{{n - 1}}} \right)}}$

26. $\int\limits_0^{ + \infty } {W\left( {\frac{1}{{x^2 }}} \right)} dx = \sqrt {2\pi }$

27. $\int\limits_0^{ + \infty } {\frac{{W\left( x \right)}}{{x\sqrt x }}} dx = 2\sqrt {2\pi }$

28. Let $\alpha ,\beta \in \Re + $. Integrate $ \int\limits_0^{ + \infty } {\left( {\exp \left( { - \theta ^\alpha } \right) - \frac{1}{{1 + \theta ^\beta }}} \right)\frac{{d\theta }}{\theta }} = - \frac{1}{\alpha }\gamma$ where $W$ is the Lambert W function

29. $\int\limits_0^{\frac{\pi }{2}} {\frac{{\ln ^2 \sin x\ln ^2 \cos x}}{{\sin x\cos x}}dx} = \frac{1}{4}\left( {2\zeta \left( 5 \right) - \zeta \left( 2 \right)\zeta \left( 3 \right)} \right)$

30. $\int\limits_0^{\frac{\pi }{2}} {4\cos ^2 x\left( {\ln \cos x} \right)^2 dx} = - \pi \ln 2 + \pi \ln ^2 2 - \frac{\pi }{2} + \frac{{\pi ^3 }}{{12}}$

31. $\int\limits_0^1 {\int\limits_0^1 {\frac{{dxdy}}{{\left( {\left[ {\frac{x}{y}} \right] + 1} \right)^2 }}} } = \frac{1}{2}\left( {\zeta \left( 3 \right) + 1 - \zeta \left( 2 \right)} \right)$

32. $\int\limits_0^1 {\int\limits_0^1 {\ln \left( {1 - xy} \right)\ln x\ln ydxdy} } = \zeta \left( 2 \right) + \zeta \left( 3 \right) + \zeta \left( 4 \right) - 4$

33. $\int\limits_0^1 {\int\limits_0^1 {...\int\limits_0^1 {\ln \left( {1 - \prod\limits_{1 \leqslant i \leqslant n} {x_i } } \right)\prod\limits_{1 \leqslant i \leqslant n} {\ln x_i } dx_1 dx_2 ...dx_n } } } = \left( { - 1} \right)^{n - 1} \left( { - 2n + \sum\limits_{1 \leqslant k \leqslant 2n} {\zeta \left( k \right)} } \right)$

34. Prove that $\int_0^{\frac{\pi }{2}} {\arctan \left( {1 - \left( {\sin x\cos x} \right)^2 } \right)} dx = \pi \left( {\frac{\pi }
{4} - \arctan \sqrt {\frac{{\sqrt 2 - 1}}{2}} } \right)$

35. Let $s>0$ and $\alpha \in \left( 0,1 \right)$. Prove that $\int\limits_{0}^{+\infty }{\frac{\text{L}{{\text{i}}_{s}}\left( -x \right)}{{{x}^{1+\alpha }}}dx}=-\frac{\pi }{{{\alpha }^{s}}\sin \left( \pi \alpha \right)}$

36. $\mathop {\lim }\limits_{n \to \infty } \int_{ - \pi }^\pi {\frac{{n!2^{2n\cos \left( \phi \right)} }}{{\left| {\prod\limits_{k = 1}^n {\left( {2ne^{i\phi } - k} \right)} } \right|}}} d\phi = 2\pi$

37. $\int\limits_{0}^{\frac{\sqrt{6}-\sqrt{2}-1}{\sqrt{6}-\sqrt{2}+1}}{\frac{\ln x}{\sqrt{x^{2}-2\left( 15+8\sqrt{3} \right)x+1}}\cdot \frac{dx}{x-1}}=\frac{2}{3}\left( 2-\sqrt{3} \right)G$ where $G$ is the Catalan's constant

38. Let $0<r<1$ and $r<s$ Prove that $\int_{-1}^{1}\frac{1}{x}\sqrt{\frac{1+x}{1-x}}\log \left | \frac{1+2rsx+\left ( r^{2}+s^{2}-1 \right )x^{2}}{1-2rsx+\left ( r^{2}+s^{2}-1 \right )x^{2}} \right |dx=4\pi \arcsin r$

39. $\int\limits_{0}^{1}{\cosh \left( \alpha \ln x \right)\ln \left( 1+x \right)\frac{dx}{x}}=\frac{1}{2\alpha }\left( \pi \csc \left( \pi \alpha \right)-\frac{1}{\alpha } \right)$

40. Let $\alpha \ne 0$ be a real number. Prove that $\int\limits_{0}^{+\infty }{\frac{\ln \tan ^{2}\left( \alpha x \right)}{1+x^{2}}dx}=\pi \ln \tanh \alpha$

41. Consider $a>0,\ b\in \Re$. Prove that $\int\limits_{ - \infty }^{ + \infty } {\frac{{a^2 }}{{\left( {e^x - ax - b} \right)^2 + \left( {a\pi } \right)^2 }}} dx = \frac{1}{{1 + W\left( {\frac{1}{a}e^{ - \frac{b}{a}} } \right)}}$

42. $\int_0^\pi {\sin \left( {n\alpha } \right)\arctan \left( {\frac{{\tan \left( {\frac{\alpha }{2}} \right)}}{{\tan \left( {\frac{\varphi }{2}} \right)}}} \right)} d\alpha = \frac{\pi }{{2n}}\left[ {\left( {\sec \left( \varphi \right) - \tan \left( \varphi \right))^n - \left( { - 1} \right)^n } \right)} \right],\left| {n \in {\Bbb Z}^ + ,0 < \varphi < \frac{\pi }{2}} \right|$

43. $\int\limits_{0}^{+\infty }{\frac{\cos \alpha x-\cos \beta x}{\sin \theta x}\frac{dx}{x}}=\log \left( \frac{\cosh \frac{\beta \pi }{2\theta }}{\cosh \frac{\alpha \pi }{2\theta }} \right)$

44. $\int\limits_0^\pi {\log \left( {1 - \cos x} \right)\log \left( {1 + \cos x} \right)dx} = \pi \log ^2 2 - \frac{{\pi ^3 }}
{6}$

45. $\int\limits_{0}^{+\infty }{\frac{\arctan x}{\sinh \left( \frac{\pi x}{2} \right)}dx}=4\log \Gamma \left( \frac{1}{4} \right)-2\log \pi -3\log 2$

46. $\int\limits_{0}^{\frac{\pi }{2}}{x\cot x\log \sin xdx}=-\frac{{{\pi }^{3}}}{48}-\frac{\pi }{4}{{\ln }^{2}}2$

47. $\int\limits_{0}^{1}{\log \left( \text{arcsech}x \right)dx}=-\gamma -2\log 2-2\log \left( \frac{\Gamma \left( \frac{3}{4} \right)}{\Gamma \left( \frac{1}{4} \right)} \right)$

48. $\int\limits_{0}^{1}{\sqrt{\frac{1-8{{x}^{2}}+16{{x}^{4}}}{1+7{{x}^{2}}-8{{x}^{4}}}}\exp \left( \frac{4x\sqrt{1-{{x}^{2}}}}{\sqrt{1+8{{x}^{2}}}} \right)dx}=e-1$

49. Let $\left| \Im \left( n \right) \right|<1$ Prove that $\int\limits_{0}^{+\infty }{\frac{\cos \left( n\pi x \right)}{\cosh \left( \pi x \right)}\cdot {{e}^{-i\pi {{x}^{2}}}}dx}=\frac{1+\sqrt{2}\sin \frac{{{n}^{2}}\pi }{4}}{2\sqrt{2}\cosh \frac{n\pi }{2}}+i\frac{1-\sqrt{2}\cos \frac{{{n}^{2}}\pi }{4}}{2\sqrt{2}\cosh \frac{n\pi }{2}}$

50. Let f be a function of class $C'\left[ 0,a \right]$. Prove that $\int\limits_0^{2a} {\int\limits_0^{\sqrt {2ax - x^2 } } {\frac{{x\left( {x^2 + y^2 } \right)}}{{\sqrt {4a^2 x^2 - \left( {x^2 + y^2 } \right)^2 } }}f'\left( y \right)dydx} } = \pi a^2 \left( {f\left( a \right) - f\left( 0 \right)} \right)$

51. $\int\limits_{0}^{+\infty }{\frac{\cos \alpha x}{x}\cdot \frac{\sinh \beta x}{\cosh \gamma x}dx}=\frac{1}{2}\log \left( \frac{\cosh \frac{\alpha \pi }{2\gamma }+\sin \frac{\beta \pi }{2\gamma }}{\cosh \frac{\alpha \pi }{2\gamma }-\sin \frac{\beta \pi }{2\gamma }} \right)\quad \left| \operatorname{Re}\left( \beta \right) \right|<\left| \operatorname{Re}\left( \gamma \right) \right|,\ \left| \operatorname{Re}\left( \beta \right) \right|+\left| \operatorname{Im}\left( \alpha \right) \right|<\left| \operatorname{Re}\left( \gamma \right) \right|$

52. $\int\limits_{0}^{+\infty }{\frac{\sin \alpha x}{x}\cdot \frac{\sinh \beta x}{sinh\gamma x}dx}=\arctan \left( \tan \frac{\beta \pi }{2\gamma }\tanh \frac{\alpha \pi }{2\gamma } \right)\quad \left| \operatorname{Re}\left( \beta \right) \right|<\left| \operatorname{Re}\left( \gamma \right) \right|,\ \left| \operatorname{Re}\left( \beta \right) \right|+\left| \operatorname{Im}\left( \alpha \right) \right|<\left| \operatorname{Re}\left( \gamma \right) \right|$

53. $\int\limits_{0}^{1}{\frac{x}{1+{{x}^{2}}}\cdot \arctan x\ln \left( 1-{{x}^{2}} \right)dx}=-\frac{{{\pi }^{3}}}{48}-\frac{\pi }{8}\ln 2+G\ln 2$

54. $\int\limits_{0}^{\frac{\pi }{2}}{\frac{\arctan \left( \alpha \sin x \right)}{\sin x}dx}=\frac{\pi }{2}{{\sinh }^{-1}}\alpha$

55. $\int\limits_{0}^{\frac{\pi }{2}}{\frac{{{x}^{2}}}{{{x}^{2}}+{{\log }^{2}}\left( 2\cos x \right)}dx}=\frac{\pi }{8}\cdot \left( 1-\gamma +\log 2\pi \right)$

56. $\int\limits_{0}^{+\infty }{\sin \left( {{x}^{2}} \right){{\ln }^{2}}xdx}=\frac{\sqrt{2\pi }}{64}\cdot {{\left( 4\ln 2+2\gamma -\pi \right)}^{2}}$

57. $\int\limits_{0}^{+\infty }{{{e}^{-\alpha x}}\sin \left( \beta x \right){{x}^{s-1}}dx}=\frac{\Gamma \left( s \right)}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}}}\cdot \sin \left( s\arctan \frac{\beta }{\alpha } \right)$

58. $\int\limits_{0}^{+\infty }{{{e}^{-\alpha x}}\cos \left( \beta x \right){{x}^{s-1}}dx}=\frac{\Gamma \left( s \right)}{\sqrt{{{\alpha }^{2}}+{{\beta }^{2}}}}\cdot \cos \left( s\arctan \frac{\beta }{\alpha } \right)$

59. $\int\limits_{0}^{+\infty }{\frac{1}{1+{{e}^{\pi x}}}\cdot \frac{x}{1+{{x}^{2}}}dx}=\frac{1}{2}\cdot \left( \log 2-\gamma \right)$

60. $\int\limits_{0}^{+\infty }{\frac{\cos \left( {{x}^{p}} \right)-{{e}^{-{{x}^{q}}}}}{{{x}^{1+r}}}dx}=\frac{\Gamma \left( 1-\frac{r}{p} \right)\Gamma \left( 1+\frac{r}{p} \right)-\Gamma \left( 1+\frac{r}{2p} \right)\Gamma \left( 1-\frac{r}{2p} \right)}{r\Gamma \left( 1+\frac{r}{p} \right)}$

61. $\int\limits_{\pi }^{+\infty }{\frac{\sin x}{x}dx}+\frac{1}{2}\int\limits_{2\pi }^{+\infty }{\frac{\sin x}{x}dx}+\frac{1}{3}\int\limits_{3\pi }^{+\infty }{\frac{\sin x}{x}dx+...}=\frac{\pi }{2}\cdot \left( 1-\ln \pi \right)$

62. $\int\limits_{0}^{+\infty }{\frac{\sin \left( \frac{\omega x}{2} \right)}{x\left( {{e}^{x}}-1 \right)}dx}=\frac{1}{4}\cdot \ln \left( \frac{\sinh \left( \pi \omega \right)}{\pi \omega } \right)$

63. $\int\limits_{0}^{+\infty }{\frac{1-\cos x}{{{x}^{2}}}{{e}^{-kx}}dx}=\arctan \frac{1}{k}-k\cdot \ln \left( \frac{\sqrt{1+{{k}^{2}}}}{k} \right)$

64. $\int\limits_{0}^{+\infty }{\sin xsin\sqrt{x}{{e}^{-\alpha x}}dx}=\frac{\sqrt{\pi }}{2}\cdot \frac{\exp \left( -\frac{\alpha }{4}\cdot \frac{1}{1+{{\alpha }^{2}}} \right)}{\sqrt[4]{{{\left( 1+{{\alpha }^{2}} \right)}^{3}}}}\cdot \sin \left( \frac{3}{2}\arctan \frac{1}{\alpha }-\frac{1}{4}\cdot \frac{1}{1+{{\alpha }^{2}}} \right)$

65. $\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{1-{{x}^{2}}}{\left( 1+{{x}^{2}}{{y}^{2}} \right){{\ln }^{2}}\left( xy \right)}dxdy}}=-2\log \left( \frac{2\Gamma \left( \frac{3}{4} \right)}{\Gamma \left( \frac{1}{4} \right)} \right)$

66. $\int\limits_{0}^{+\infty }{\sin \left( \frac{1}{{{x}^{2}}} \right){{e}^{-\alpha {{x}^{2}}}}dx}=\frac{1}{2}\sqrt{\frac{\pi }{\alpha }}{{e}^{-\sqrt{2\alpha }}}\sin \sqrt{2\alpha }$

67. $\int\limits_{0}^{1}{\frac{\ln \left( {{x}^{2}} \right)}{\left( 1+{{x}^{2}} \right)\left( {{\pi }^{2}}+{{\ln }^{2}}x \right)}dx}=\ln 2-\frac{1}{2}$

68. $\int\limits_{0}^{1}{\frac{\ln \left( {{\pi }^{2}}+{{\ln }^{2}}x \right)}{1+{{x}^{2}}}dx}=\pi \ln \left( \frac{1}{2}\sqrt{\frac{\pi }{2}}\cdot \frac{\Gamma \left( \frac{1}{4} \right)}{\Gamma \left( \frac{3}{4} \right)} \right)$

69. $\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{{{x}^{2}}-1}{\left( 1+{{x}^{2}}{{y}^{2}} \right){{\ln }^{2}}\left( xy \right)}dx}dx}=\frac{1}{2}-\frac{2C}{\pi }+\ln \left( \frac{2\sqrt{2}\pi }{{{\Gamma }^{2}}\left( \frac{1}{4} \right)} \right)$

70. $\int\limits_{0}^{+\infty }{\frac{dx}{\left( {{x}^{2}}+\frac{{{\pi }^{2}}}{4} \right)\cosh x}}=\frac{2\ln 2}{\pi }$

71. $\int\limits_{0}^{1}{{{\left( tan{{h}^{-1}}x \right)}^{z}}dx}=\frac{\zeta \left( z \right)}{{{2}^{2z-1}}}\cdot \Gamma \left( z+1 \right)\left( {{2}^{z}}-2 \right)\quad z\in \mathbb{N},z\ge 2$

72. $\int\limits_{0}^{+\infty }{x{{e}^{-x}}{{\left( \int\limits_{0}^{\frac{\pi }{2}}{\left( 1-{{e}^{x-x\csc t}} \right){{\sec }^{2}}tdt} \right)}^{2}}dx}=\frac{1}{3}$

73. $\int\limits_{0}^{+\infty }{{{e}^{-x}}\ln \ln \left( {{e}^{x}}+\sqrt{{{e}^{2x}}-1} \right)dx}=-\gamma +4\log \Gamma \left( \frac{1}{4} \right)-3\log 2-2\log \pi$

74. $\int\limits_{0}^{1}{\int\limits_{0}^{1}{\int\limits_{0}^{1}{\left\{ \frac{x}{y} \right\}\left\{ \frac{y}{z} \right\}\left\{ \frac{z}{x} \right\}dxdydz}}}=1+\frac{\zeta \left( 2 \right)\zeta \left( 3 \right)}{6}-\frac{3\zeta \left( 2 \right)}{4}$

75. $\int\limits_{0}^{\pi }{x\cot \left( \frac{x}{4} \right)dx}=2\pi \log 2+8C$

76. $\int\limits_{0}^{\frac{\pi }{2}}{x{{2}^{s}}co{{s}^{s}}x\sin \left( sx \right)dx}=\frac{\pi }{4}\cdot \left( \gamma +\psi \left( s+1 \right) \right)$

77. $\int\limits_{0}^{+\infty }{{{x}^{s-1}}{{\left( \arctan x \right)}^{2}}dx}=\frac{\pi }{2s\sin \frac{\pi s}{2}}\cdot \left( \gamma +\psi \left( \frac{1-s}{2} \right)+2\log 2 \right)$

78. $\int\limits_{0}^{\frac{\pi }{2}}{xta{{n}^{s}}xdx}=\frac{\pi }{4\sin \frac{\pi s}{2}}\cdot \left( \psi \left( \frac{1}{2} \right)-\psi \left( \frac{1-s}{2} \right) \right)$

79. $\int\limits_{0}^{+\infty }{\frac{\exp \left( -{{x}^{2}} \right)}{{{\left( {{x}^{2}}+\frac{1}{2} \right)}^{2}}}dx}=\sqrt{\pi }$

80. $\int\limits_{0}^{+\infty }{\frac{1}{x}\left( \frac{\sinh \alpha x}{\sinh x}-\alpha {{e}^{-2x}} \right)dx}=\log \left( \frac{\pi }{\cos \frac{\alpha \pi }{2}{{\Gamma }^{2}}\left( \frac{\alpha +1}{2} \right)} \right)\quad \left| \alpha \right|<1$

81. $\int\limits_{0}^{+\infty }{\frac{\ln \left( {{x}^{2}}+{{\alpha }^{2}} \right)}{\cosh x+\cos t}dx}=\frac{2\pi }{\sin t}\log \left( \frac{\Gamma \left( \frac{\alpha }{2\pi }+\frac{\pi +t}{2\pi } \right)}{\Gamma \left( \frac{\alpha }{2\pi }+\frac{\pi -t}{2\pi } \right)} \right)+\frac{2t}{\sin t}\ln 2\pi$

82. $\int\limits_{0}^{+\infty }{\frac{{{\left( \sinh \left( sx \right) \right)}^{2}}}{x{{\left( {{e}^{x}}-1 \right)}^{3}}}dx}=\log \left( \frac{2\pi s}{\sin \left( 2\pi s \right)} \right)\quad 0<s<\frac{1}{2}$

83. $\int\limits_{0}^{+\infty }{\frac{{{x}^{s-1}}\sinh \left( \pi x \right)}{{{\left( \cosh \left( \pi x \right)-1 \right)}^{3}}}dx}=\frac{\Gamma \left( s \right)}{3{{\pi }^{s}}}\cdot \left( \zeta \left( 4-s \right)-\zeta \left( 2-s \right) \right)$

84. $\int\limits_{0}^{1}{\int\limits_{0}^{1}{\int\limits_{0}^{1}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}dxdydz}}}=\log \left( \sqrt{3}+1 \right)-\frac{\log 2}{2}+\frac{\sqrt{3}}{4}-\frac{\pi }{24}$

85. $\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{{{x}^{\alpha -1}}{{y}^{\beta -1}}}{\left( 1+xy \right)\log \left( xy \right)}dxdy}}=\frac{1}{\alpha -\beta }\cdot \log \left( \frac{\Gamma \left( \frac{\alpha }{2} \right)\Gamma \left( \frac{1}{2}+\frac{\beta }{2} \right)}{\Gamma \left( \frac{\beta }{2} \right)\Gamma \left( \frac{1}{2}+\frac{\alpha }{2} \right)} \right)$

86. $\int\limits_{-\infty }^{+\infty }{\frac{1}{1+\frac{{{x}^{2}}}{{{\alpha }^{2}}}}\cdot \prod\limits_{k=1}^{+\infty }{\frac{1+\frac{{{x}^{2}}}{{{\left( \beta +k \right)}^{2}}}}{1+\frac{{{x}^{2}}}{{{\left( \alpha +k \right)}^{2}}}}dx}}=\sqrt{\pi }\cdot \frac{\Gamma \left( \beta +1 \right)}{\Gamma \left( \alpha \right)}\cdot \frac{\Gamma \left( \alpha +\frac{1}{2} \right)}{\Gamma \left( \beta +\frac{1}{2} \right)}\cdot \frac{\Gamma \left( \beta -\alpha +\frac{1}{2} \right)}{\Gamma \left( \beta -\alpha +1 \right)}\quad 0<\alpha <\beta +\frac{1}{2}$

87. $\int\limits_{0}^{+\infty }{\frac{1}{x}\left( \frac{\sinh \left( ax \right)}{\sinh x}-a{{e}^{-2x}} \right)dx}=\log \left( \frac{\pi }{\cos \left( \frac{a\pi }{2} \right){{\Gamma }^{2}}\left( \frac{a+1}{2} \right)} \right)$

88. $\int\limits_{0}^{+\infty }{{{x}^{2}}{{e}^{-{{x}^{2}}}}erf\left( x \right)\log xdx}=\frac{2-\log 2}{16}\sqrt{\pi }-\frac{\gamma +\log 2}{16\sqrt{\pi }}\left( \pi +2 \right)+\frac{G}{4\sqrt{\pi }}$

89. $\int\limits_{0}^{\frac{\pi }{2}}{\sin \left( 2nx \right)\sinh \left( a\sin x \right)\sin \left( a\cos x \right)dx}={{\left( -1 \right)}^{n+1}}\frac{\pi }{4}\cdot \frac{{{a}^{2n}}}{\left( 2n \right)!}$

90. Let $\beta >0$ and $\alpha \in \left( -\frac{\pi }{2},\frac{\pi }{2} \right)$. Prove that $\int\limits_{0}^{+\infty }{{{e}^{-t\cos \alpha }}{{t}^{\beta -1}}\cos \left( t\sin \alpha \right)dx}=\Gamma \left( \beta \right)\cos \left( \beta \sin \alpha \right)$

91. $\int\limits_{0}^{+\infty }{\frac{\ln \left( 1+x \right)\ln \left( 1+\frac{1}{{{x}^{2}}} \right)}{x}dx}=\pi G-\frac{3}{8}\zeta \left( 3 \right)$

92. $\int\limits_{-\infty }^{+\infty }{\int\limits_{-\infty }^{+\infty }{\text{sign}\left( x \right)\text{sign}\left( y \right){{e}^{-\frac{{{x}^{2}}+{{y}^{2}}}{2}}}\sin \left( xy \right)dxdy}=2\sqrt{2}\log \left( 1+\sqrt{2} \right)}$

93. $\int\limits_{0}^{+\infty }{\left( \frac{x}{{{\log }^{2}}\left( {{e}^{{{x}^{2}}}}-1 \right)}-\frac{x}{\sqrt{{{e}^{{{x}^{2}}}}-1}{{\log }^{2}}\left( {{e}^{{{x}^{2}}}}-1 \right)}-\frac{x}{\sqrt{{{e}^{{{x}^{2}}}}-1}\log \left( {{\left( {{e}^{{{x}^{2}}}}-1 \right)}^{2}} \right)} \right)dx}=\frac{G}{\pi }$

94. $\int\limits_{0}^{1}{{{B}_{2n+1}}\left( x \right)\cot \left( \pi x \right)dx}=\frac{2\left( 2n+1 \right)!}{{{\left( -1 \right)}^{n+1}}{{\left( 2\pi \right)}^{2n+1}}}\zeta \left( 2n+1 \right)$ where ${{B}_{2n+1}}\left( x \right)$ is the Bernoulli Polynomial

95. $\int\limits_{0}^{+\infty }{\frac{x}{1+{{x}^{4}}}\arctan \left( \frac{p\sin qx}{1+p\cos qx} \right)dx}=\frac{\pi }{2}\arctan \left( \frac{p\sin \left( \frac{q}{\sqrt{2}} \right)}{{{e}^{\frac{q}{\sqrt{2}}}}+p\cos \left( \frac{q}{\sqrt{2}} \right)} \right)$

96. Let $m\in \Re$ and $a\in \left( -1,1 \right)$ Calculate
$\int\limits_{0}^{2\pi }{\frac{{{e}^{m\cos \theta }}\left( \cos \left( m\sin \theta \right)-a\sin \left( \theta +m\sin \theta \right) \right)}{1-2a\sin \theta +{{a}^{2}}}d\theta }$

97. Prove that $\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{{{\left( xy \right)}^{s-1}}{{y}^{n}}}{\left( 1-xy \right)\log \left( xy \right)}dxdy}}=\frac{\Gamma '\left( s \right)}{\Gamma \left( s \right)}-\frac{\log \left( n! \right)}{n}$

98. $\int\limits_{0}^{+\infty }{\sin \left( nx \right)\left( \cot x+\coth x \right){{e}^{-nx}}dx}=\frac{\pi }{2}\cdot \frac{\sinh \left( n\pi \right)}{\cosh \left( n\pi \right)-\cos \left( n\pi \right)}$

99. $\int\limits_{0}^{\frac{\pi }{3}}{{{\log }^{2}}\left( \frac{\sin x}{\sin \left( x+\frac{\pi }{3} \right)} \right)dx}=\frac{5{{\pi }^{3}}}{81}$

100. $\int\limits_{0}^{2\pi }{{{x}^{2}}\log \left( 1-\exp \left( ix \right) \right)dx}=2\pi \zeta \left( 4 \right)-8i{{\pi }^{2}}\zeta \left( 3 \right)$

100+1. Let $a\in \left( 0,1 \right)$ Prove that $\int\limits_{0}^{1}{\frac{\log \log \frac{1}{x}}{1+2x\cos \left( a\pi \right)+{{x}^{2}}}dx}=\frac{\pi }{2\sin \left( a\pi \right)}\left( a\log \left( 2\pi \right)+\log \frac{\Gamma \left( \frac{1}{2}+\frac{a}{2} \right)}{\Gamma \left( \frac{1}{2}-\frac{a}{2} \right)} \right)$

Bonus
1. $\int\limits_{0}^{+\infty }{\frac{\cos x}{x}{{\left( \int\limits_{0}^{x}{\frac{\sin t}{t}dt} \right)}^{2}}dx}=-\frac{7}{6}\zeta \left( 3 \right)$

2. $\int\limits_{0}^{+\infty }{\frac{{{x}^{a-1}}\sin x}{\cos x+\cosh x}dx}={{2}^{1-\frac{a}{2}}}\Gamma \left( a \right)\sin \left( \frac{a\pi }{4} \right)\left( 1-{{2}^{1-a}} \right)\zeta \left( a \right)$

3. $\int\limits_{0}^{+\infty }{\frac{\cos \left( tx \right)}{\left( 1+{{x}^{2}} \right)\cosh \left( \frac{\pi x}{2} \right)}dx}=\cosh t\log \left( 2\cosh t \right)-t\sinh t$

4. $\int\limits_{0}^{+\infty }{\frac{{{t}^{s-1}}}{{{z}^{-1}}{{e}^{t}}-1}dt}=\Gamma \left( s \right)\text{L}{{\text{i}}_{s}}\left( z \right)$

5. $\int\limits_{0}^{+\infty }{\exp \left( -2u \right)\left( \frac{1}{u\sinh u}-\frac{1}{{{u}^{2}}coshu} \right)du}=2-\log 2-\frac{4G}{\pi }$

6. $\int\limits_{0}^{+\infty }{\frac{\sin \left( bt \right)}{t}{{e}^{-at}}\ln tdt}=-\left( \gamma +\frac{\ln \left( {{a}^{2}}+{{b}^{2}} \right)}{2} \right)\arctan \frac{b}{a}$

7. $\int\limits_{0}^{1}{\frac{\left( a-t \right)\ln \left( 1-t \right)}{1-2at+{{t}^{2}}}dt}=\frac{{{\pi }^{2}}}{12}-\frac{{{\left( \arccos a-\pi \right)}^{2}}}{8}-\frac{{{\ln }^{2}}\left( 2-2a \right)}{8}$

8. $\int\limits_{0}^{1}{{{\left\{ \frac{1}{x} \right\}}^{2}}dx}=\ln \left( 2\pi \right)-1-\gamma$

9. $\int\limits_{0}^{1}{{{\left\{ \frac{1}{x} \right\}}^{2}}\left\{ \frac{1}{1-x} \right\}dx}=2+\gamma -\ln \left( 4\pi \right)$

10. $\int\limits_{0}^{1}{{{\left\{ \frac{x}{y} \right\}}^{2}}dxdy}=\frac{1}{2}\ln \left( 2\pi \right)-\frac{1}{3}-\frac{\gamma }{2}$