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 $int_0^{frac{pi }{2}} {arccos left( {frac{{cos heta }}{{1 + 2cos heta }}} ight)d heta = frac{5}{{24}}pi ^2 }$
2. $int_0^{frac{pi }{2}} {arccos left( {frac{1}{{1 + 2cos heta }}} ight)d heta = frac{1}{8}pi ^2 }$
3. $int_0^{frac{pi }{2}} {arccos left( {frac{{1 - cos heta }}{{2cos heta }}} ight)d heta = frac{{11}}{{72}}pi ^2 }$
4. For any $n$ natural number. Show that $intlimits_{0}^{2pi }{frac{left( 1+2cos x ight)^{n}cos nx}{3+2cos x}dx}=frac{2pi }{sqrt{5}}left( 3-sqrt{5} ight)^{n}$
5. Let $0<a<1$ Prove that $intlimits_{0}^{2pi }{frac{cos ^{2}3x}{1+a^{2}-2acos 2x}dx}=frac{a^{2}-a+1}{1-a}pi$
6. For $a>1$ Prove that $intlimits_{-pi }^{pi }{frac{xsin x}{1+a^{2}-2acos x}dx}=frac{pi }{2}ln left( 1+frac{1}{a} ight)$
7. $intlimits_{0}^{1}{frac{ln ln frac{1}{x}}{left( 1+x ight)^{2}}dx}=frac{1}{2}left( ln pi -ln 2-gamma ight)$
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 $displaystylefrac{1}{2pi}int_0^{2pi}log|z-e^{i heta}|\,d heta = left{ egin{array}{ll} 0 & ext{ si }|z|<1\ log|z| & ext{ si }|z|ge1 end{array} ight.$
10. $intlimits_{0}^{+infty }{exp left( -a^{2}xleft( frac{x-6}{x-2} ight)^{2} ight)frac{dx}{sqrt{x}}}=frac{sqrt{pi }}{a}$
11. Let $alpha >0$ Prove that $Ileft( alpha ight)=intlimits_{0}^{frac{pi }{2}}{arctan left( frac{2alpha sin ^{2}x}{alpha ^{2}-1+cos ^{2}x} ight)dx}=pi arctan left( frac{1}{2alpha } ight)$
12. $intlimits_0^1 {frac{{log left( {1 - x} ight)}}{x} cdot frac{{2z}}{{log ^2 x + left( {2pi z} ight)^2 }}dx} = - log left( {frac{{z!e^z }}{{z^z sqrt {2pi z} }}} ight),;;operatorname{Re} left( z ight) > 0$
13.$intlimits_{0}^{1}{frac{1-x}{log x}cdot left( x+x^{2}+x^{2^{2}}+... ight)dx}$
14. Let $a_k > 0$ and $a_0 > sumlimits_{k = 1}^n {a_k }$. Show that $intlimits_0^{ + infty } {prodlimits_{k = 0}^n {frac{{sin left( {a_k x} ight)}}{x}dx} } = frac{pi }{2}prodlimits_{k = 1}^n {a_k }$
15. Let $0 < z < 1,alpha > 0,eta in {Bbb C}$
$intlimits_0^{ + infty } {sin left( {alpha t^{frac{1}{z}} + eta }
ight)} dt = frac{{Gamma left( {z + 1}
ight)}}
{{alpha ^z }}sin left( {frac{{pi z}}{2} + eta }
ight)$
16. $operatorname{Re}left( alpha
ight)ge 1$
$intlimits_{ - infty }^{ + infty } {left| {sin x}
ight|^{alpha - 1} frac{{sin x}}{x}} dx = 2^{alpha - 1} frac{{Gamma ^2 left( {frac{alpha }{2}}
ight)}}{{Gamma left( alpha
ight)}}$
17. $intlimits_0^1 {sin left( {pi x} ight)} x^x left( {1 - x} ight)^{1 - x} dx = frac{{pi e}}{{24}}$
18. $intlimits_0^{frac{pi }{4}} {frac{{x^3 }}{{sin ^2 x}}} dx = frac{{3pi }}{4}G - frac{{pi ^3 }}{{64}} + frac{{3pi ^2 }}
{{32}}log 2 - frac{{105}}{{64}}varsigma left( 3
ight)$
19. Let $ heta > 0$
$intlimits_{ - infty }^{ + infty } {frac{{left| {cos heta x}
ight|}}{{1 + x^2 }}dx} = 4cosh heta arctan e^{ - heta }$
20. Let $alpha geqslant 0, heta in {Bbb C}ackslash pi {Bbb Z}$
$intlimits_{ - infty }^{ + infty } {frac{{cos alpha x}}{{1 + 2cos heta x + x^2 }}dx} = frac{pi }{{sin heta }}frac{{cos left( {alpha cos heta }
ight)}}{{e^{alpha sin heta } }}$
21. Show that $int_{0}^{frac{pi }{2}}{frac{d heta }{1+sin ^{2} an heta }}=frac{pi }{2sqrt{2}}left( frac{e^{2}+3-2sqrt{2}}{e^{2}-3+2sqrt{2}} ight)$
22. Let $ heta in left[ 0,frac{pi }{2} ight)$ Prove that $intlimits_{-infty }^{infty }{frac{arctan x}{x^{2}-2xsin heta +1}dx}$
23. Given the function $yleft(x ight):left [0,1 ight] oleft [0,1 ight]$ continuous and decreasing such that $x^{a}-x^{b} = y^{a}-y^{b}$. Compute $intlimits_{0}^{1}{frac{ln left( yleft( x ight) ight)}{x}dx}$
24. $int_{0}^{1}left ( -1 ight )^{left [ 1994x ight ] + left [ 1995x ight ]}inom{1993}{left [ 1994x ight ]}inom{1994}{left [ 1995x ight ]}dx$
25. $intlimits_0^1 {frac{{dx}}{{1 + {}_2F_1 left( {frac{1}{n},x;frac{1}{n};frac{1}{n}} ight)}}} = frac{{log left( {frac{{2n}}{{2n - 1}}} ight)}}{{log left( {frac{n}{{n - 1}}} ight)}}$
26. $intlimits_0^{ + infty } {Wleft( {frac{1}{{x^2 }}} ight)} dx = sqrt {2pi }$
27. $intlimits_0^{ + infty } {frac{{Wleft( x ight)}}{{xsqrt x }}} dx = 2sqrt {2pi }$
28. Let $alpha ,eta in Re + $. Integrate $ intlimits_0^{ + infty } {left( {exp left( { - heta ^alpha } ight) - frac{1}{{1 + heta ^eta }}} ight)frac{{d heta }}{ heta }} = - frac{1}{alpha }gamma$ where $W$ is the Lambert W function
29. $intlimits_0^{frac{pi }{2}} {frac{{ln ^2 sin xln ^2 cos x}}{{sin xcos x}}dx} = frac{1}{4}left( {2zeta left( 5 ight) - zeta left( 2 ight)zeta left( 3 ight)} ight)$
30. $intlimits_0^{frac{pi }{2}} {4cos ^2 xleft( {ln cos x} ight)^2 dx} = - pi ln 2 + pi ln ^2 2 - frac{pi }{2} + frac{{pi ^3 }}{{12}}$
31. $intlimits_0^1 {intlimits_0^1 {frac{{dxdy}}{{left( {left[ {frac{x}{y}} ight] + 1} ight)^2 }}} } = frac{1}{2}left( {zeta left( 3 ight) + 1 - zeta left( 2 ight)} ight)$
32. $intlimits_0^1 {intlimits_0^1 {ln left( {1 - xy} ight)ln xln ydxdy} } = zeta left( 2 ight) + zeta left( 3 ight) + zeta left( 4 ight) - 4$
33. $intlimits_0^1 {intlimits_0^1 {...intlimits_0^1 {ln left( {1 - prodlimits_{1 leqslant i leqslant n} {x_i } } ight)prodlimits_{1 leqslant i leqslant n} {ln x_i } dx_1 dx_2 ...dx_n } } } = left( { - 1} ight)^{n - 1} left( { - 2n + sumlimits_{1 leqslant k leqslant 2n} {zeta left( k ight)} } ight)$
34. Prove that $int_0^{frac{pi }{2}} {arctan left( {1 - left( {sin xcos x}
ight)^2 }
ight)} dx = pi left( {frac{pi }
{4} - arctan sqrt {frac{{sqrt 2 - 1}}{2}} }
ight)$
35. Let $s>0$ and $alpha in left( 0,1 ight)$. Prove that $intlimits_{0}^{+infty }{frac{ ext{L}{{ ext{i}}_{s}}left( -x ight)}{{{x}^{1+alpha }}}dx}=-frac{pi }{{{alpha }^{s}}sin left( pi alpha ight)}$
36. $mathop {lim }limits_{n o infty } int_{ - pi }^pi {frac{{n!2^{2ncos left( phi ight)} }}{{left| {prodlimits_{k = 1}^n {left( {2ne^{iphi } - k} ight)} } ight|}}} dphi = 2pi$
37. $intlimits_{0}^{frac{sqrt{6}-sqrt{2}-1}{sqrt{6}-sqrt{2}+1}}{frac{ln x}{sqrt{x^{2}-2left( 15+8sqrt{3} ight)x+1}}cdot frac{dx}{x-1}}=frac{2}{3}left( 2-sqrt{3} ight)G$ where $G$ is the Catalan's constant
38. Let $0<r<1$ and $r<s$ Prove that $int_{-1}^{1}frac{1}{x}sqrt{frac{1+x}{1-x}}log left | frac{1+2rsx+left ( r^{2}+s^{2}-1 ight )x^{2}}{1-2rsx+left ( r^{2}+s^{2}-1 ight )x^{2}} ight |dx=4pi arcsin r$
39. $intlimits_{0}^{1}{cosh left( alpha ln x ight)ln left( 1+x ight)frac{dx}{x}}=frac{1}{2alpha }left( pi csc left( pi alpha ight)-frac{1}{alpha } ight)$
40. Let $alpha e 0$ be a real number. Prove that $intlimits_{0}^{+infty }{frac{ln an ^{2}left( alpha x ight)}{1+x^{2}}dx}=pi ln anh alpha$
41. Consider $a>0, bin Re$. Prove that $intlimits_{ - infty }^{ + infty } {frac{{a^2 }}{{left( {e^x - ax - b} ight)^2 + left( {api } ight)^2 }}} dx = frac{1}{{1 + Wleft( {frac{1}{a}e^{ - frac{b}{a}} } ight)}}$
42. $int_0^pi {sin left( {nalpha } ight)arctan left( {frac{{ an left( {frac{alpha }{2}} ight)}}{{ an left( {frac{varphi }{2}} ight)}}} ight)} dalpha = frac{pi }{{2n}}left[ {left( {sec left( varphi ight) - an left( varphi ight))^n - left( { - 1} ight)^n } ight)} ight],left| {n in {Bbb Z}^ + ,0 < varphi < frac{pi }{2}} ight|$
43. $intlimits_{0}^{+infty }{frac{cos alpha x-cos eta x}{sin heta x}frac{dx}{x}}=log left( frac{cosh frac{eta pi }{2 heta }}{cosh frac{alpha pi }{2 heta }} ight)$
44. $intlimits_0^pi {log left( {1 - cos x}
ight)log left( {1 + cos x}
ight)dx} = pi log ^2 2 - frac{{pi ^3 }}
{6}$
45. $intlimits_{0}^{+infty }{frac{arctan x}{sinh left( frac{pi x}{2} ight)}dx}=4log Gamma left( frac{1}{4} ight)-2log pi -3log 2$
46. $intlimits_{0}^{frac{pi }{2}}{xcot xlog sin xdx}=-frac{{{pi }^{3}}}{48}-frac{pi }{4}{{ln }^{2}}2$
47. $intlimits_{0}^{1}{log left( ext{arcsech}x ight)dx}=-gamma -2log 2-2log left( frac{Gamma left( frac{3}{4} ight)}{Gamma left( frac{1}{4} ight)} ight)$
48. $intlimits_{0}^{1}{sqrt{frac{1-8{{x}^{2}}+16{{x}^{4}}}{1+7{{x}^{2}}-8{{x}^{4}}}}exp left( frac{4xsqrt{1-{{x}^{2}}}}{sqrt{1+8{{x}^{2}}}} ight)dx}=e-1$
49. Let $left| Im left( n ight) ight|<1$ Prove that $intlimits_{0}^{+infty }{frac{cos left( npi x ight)}{cosh left( pi x ight)}cdot {{e}^{-ipi {{x}^{2}}}}dx}=frac{1+sqrt{2}sin frac{{{n}^{2}}pi }{4}}{2sqrt{2}cosh frac{npi }{2}}+ifrac{1-sqrt{2}cos frac{{{n}^{2}}pi }{4}}{2sqrt{2}cosh frac{npi }{2}}$
50. Let f be a function of class $C'left[ 0,a ight]$. Prove that $intlimits_0^{2a} {intlimits_0^{sqrt {2ax - x^2 } } {frac{{xleft( {x^2 + y^2 } ight)}}{{sqrt {4a^2 x^2 - left( {x^2 + y^2 } ight)^2 } }}f'left( y ight)dydx} } = pi a^2 left( {fleft( a ight) - fleft( 0 ight)} ight)$
51. $intlimits_{0}^{+infty }{frac{cos alpha x}{x}cdot frac{sinh eta x}{cosh gamma x}dx}=frac{1}{2}log left( frac{cosh frac{alpha pi }{2gamma }+sin frac{eta pi }{2gamma }}{cosh frac{alpha pi }{2gamma }-sin frac{eta pi }{2gamma }} ight)quad left| operatorname{Re}left( eta ight) ight|<left| operatorname{Re}left( gamma ight) ight|, left| operatorname{Re}left( eta ight) ight|+left| operatorname{Im}left( alpha ight) ight|<left| operatorname{Re}left( gamma ight) ight|$
52. $intlimits_{0}^{+infty }{frac{sin alpha x}{x}cdot frac{sinh eta x}{sinhgamma x}dx}=arctan left( an frac{eta pi }{2gamma } anh frac{alpha pi }{2gamma } ight)quad left| operatorname{Re}left( eta ight) ight|<left| operatorname{Re}left( gamma ight) ight|, left| operatorname{Re}left( eta ight) ight|+left| operatorname{Im}left( alpha ight) ight|<left| operatorname{Re}left( gamma ight) ight|$
53. $intlimits_{0}^{1}{frac{x}{1+{{x}^{2}}}cdot arctan xln left( 1-{{x}^{2}} ight)dx}=-frac{{{pi }^{3}}}{48}-frac{pi }{8}ln 2+Gln 2$
54. $intlimits_{0}^{frac{pi }{2}}{frac{arctan left( alpha sin x ight)}{sin x}dx}=frac{pi }{2}{{sinh }^{-1}}alpha$
55. $intlimits_{0}^{frac{pi }{2}}{frac{{{x}^{2}}}{{{x}^{2}}+{{log }^{2}}left( 2cos x ight)}dx}=frac{pi }{8}cdot left( 1-gamma +log 2pi ight)$
56. $intlimits_{0}^{+infty }{sin left( {{x}^{2}} ight){{ln }^{2}}xdx}=frac{sqrt{2pi }}{64}cdot {{left( 4ln 2+2gamma -pi ight)}^{2}}$
57. $intlimits_{0}^{+infty }{{{e}^{-alpha x}}sin left( eta x ight){{x}^{s-1}}dx}=frac{Gamma left( s ight)}{sqrt{{{alpha }^{2}}+{{eta }^{2}}}}cdot sin left( sarctan frac{eta }{alpha } ight)$
58. $intlimits_{0}^{+infty }{{{e}^{-alpha x}}cos left( eta x ight){{x}^{s-1}}dx}=frac{Gamma left( s ight)}{sqrt{{{alpha }^{2}}+{{eta }^{2}}}}cdot cos left( sarctan frac{eta }{alpha } ight)$
59. $intlimits_{0}^{+infty }{frac{1}{1+{{e}^{pi x}}}cdot frac{x}{1+{{x}^{2}}}dx}=frac{1}{2}cdot left( log 2-gamma ight)$
60. $intlimits_{0}^{+infty }{frac{cos left( {{x}^{p}} ight)-{{e}^{-{{x}^{q}}}}}{{{x}^{1+r}}}dx}=frac{Gamma left( 1-frac{r}{p} ight)Gamma left( 1+frac{r}{p} ight)-Gamma left( 1+frac{r}{2p} ight)Gamma left( 1-frac{r}{2p} ight)}{rGamma left( 1+frac{r}{p} ight)}$
61. $intlimits_{pi }^{+infty }{frac{sin x}{x}dx}+frac{1}{2}intlimits_{2pi }^{+infty }{frac{sin x}{x}dx}+frac{1}{3}intlimits_{3pi }^{+infty }{frac{sin x}{x}dx+...}=frac{pi }{2}cdot left( 1-ln pi ight)$
62. $intlimits_{0}^{+infty }{frac{sin left( frac{omega x}{2} ight)}{xleft( {{e}^{x}}-1 ight)}dx}=frac{1}{4}cdot ln left( frac{sinh left( pi omega ight)}{pi omega } ight)$
63. $intlimits_{0}^{+infty }{frac{1-cos x}{{{x}^{2}}}{{e}^{-kx}}dx}=arctan frac{1}{k}-kcdot ln left( frac{sqrt{1+{{k}^{2}}}}{k} ight)$
64. $intlimits_{0}^{+infty }{sin xsinsqrt{x}{{e}^{-alpha x}}dx}=frac{sqrt{pi }}{2}cdot frac{exp left( -frac{alpha }{4}cdot frac{1}{1+{{alpha }^{2}}} ight)}{sqrt[4]{{{left( 1+{{alpha }^{2}} ight)}^{3}}}}cdot sin left( frac{3}{2}arctan frac{1}{alpha }-frac{1}{4}cdot frac{1}{1+{{alpha }^{2}}} ight)$
65. $intlimits_{0}^{1}{intlimits_{0}^{1}{frac{1-{{x}^{2}}}{left( 1+{{x}^{2}}{{y}^{2}} ight){{ln }^{2}}left( xy ight)}dxdy}}=-2log left( frac{2Gamma left( frac{3}{4} ight)}{Gamma left( frac{1}{4} ight)} ight)$
66. $intlimits_{0}^{+infty }{sin left( frac{1}{{{x}^{2}}} ight){{e}^{-alpha {{x}^{2}}}}dx}=frac{1}{2}sqrt{frac{pi }{alpha }}{{e}^{-sqrt{2alpha }}}sin sqrt{2alpha }$
67. $intlimits_{0}^{1}{frac{ln left( {{x}^{2}} ight)}{left( 1+{{x}^{2}} ight)left( {{pi }^{2}}+{{ln }^{2}}x ight)}dx}=ln 2-frac{1}{2}$
68. $intlimits_{0}^{1}{frac{ln left( {{pi }^{2}}+{{ln }^{2}}x ight)}{1+{{x}^{2}}}dx}=pi ln left( frac{1}{2}sqrt{frac{pi }{2}}cdot frac{Gamma left( frac{1}{4} ight)}{Gamma left( frac{3}{4} ight)} ight)$
69. $intlimits_{0}^{1}{intlimits_{0}^{1}{frac{{{x}^{2}}-1}{left( 1+{{x}^{2}}{{y}^{2}} ight){{ln }^{2}}left( xy ight)}dx}dx}=frac{1}{2}-frac{2C}{pi }+ln left( frac{2sqrt{2}pi }{{{Gamma }^{2}}left( frac{1}{4} ight)} ight)$
70. $intlimits_{0}^{+infty }{frac{dx}{left( {{x}^{2}}+frac{{{pi }^{2}}}{4} ight)cosh x}}=frac{2ln 2}{pi }$
71. $intlimits_{0}^{1}{{{left( tan{{h}^{-1}}x ight)}^{z}}dx}=frac{zeta left( z ight)}{{{2}^{2z-1}}}cdot Gamma left( z+1 ight)left( {{2}^{z}}-2 ight)quad zin mathbb{N},zge 2$
72. $intlimits_{0}^{+infty }{x{{e}^{-x}}{{left( intlimits_{0}^{frac{pi }{2}}{left( 1-{{e}^{x-xcsc t}} ight){{sec }^{2}}tdt} ight)}^{2}}dx}=frac{1}{3}$
73. $intlimits_{0}^{+infty }{{{e}^{-x}}ln ln left( {{e}^{x}}+sqrt{{{e}^{2x}}-1} ight)dx}=-gamma +4log Gamma left( frac{1}{4} ight)-3log 2-2log pi$
74. $intlimits_{0}^{1}{intlimits_{0}^{1}{intlimits_{0}^{1}{left{ frac{x}{y} ight}left{ frac{y}{z} ight}left{ frac{z}{x} ight}dxdydz}}}=1+frac{zeta left( 2 ight)zeta left( 3 ight)}{6}-frac{3zeta left( 2 ight)}{4}$
75. $intlimits_{0}^{pi }{xcot left( frac{x}{4} ight)dx}=2pi log 2+8C$
76. $intlimits_{0}^{frac{pi }{2}}{x{{2}^{s}}co{{s}^{s}}xsin left( sx ight)dx}=frac{pi }{4}cdot left( gamma +psi left( s+1 ight) ight)$
77. $intlimits_{0}^{+infty }{{{x}^{s-1}}{{left( arctan x ight)}^{2}}dx}=frac{pi }{2ssin frac{pi s}{2}}cdot left( gamma +psi left( frac{1-s}{2} ight)+2log 2 ight)$
78. $intlimits_{0}^{frac{pi }{2}}{xta{{n}^{s}}xdx}=frac{pi }{4sin frac{pi s}{2}}cdot left( psi left( frac{1}{2} ight)-psi left( frac{1-s}{2} ight) ight)$
79. $intlimits_{0}^{+infty }{frac{exp left( -{{x}^{2}} ight)}{{{left( {{x}^{2}}+frac{1}{2} ight)}^{2}}}dx}=sqrt{pi }$
80. $intlimits_{0}^{+infty }{frac{1}{x}left( frac{sinh alpha x}{sinh x}-alpha {{e}^{-2x}} ight)dx}=log left( frac{pi }{cos frac{alpha pi }{2}{{Gamma }^{2}}left( frac{alpha +1}{2} ight)} ight)quad left| alpha ight|<1$
81. $intlimits_{0}^{+infty }{frac{ln left( {{x}^{2}}+{{alpha }^{2}} ight)}{cosh x+cos t}dx}=frac{2pi }{sin t}log left( frac{Gamma left( frac{alpha }{2pi }+frac{pi +t}{2pi } ight)}{Gamma left( frac{alpha }{2pi }+frac{pi -t}{2pi } ight)} ight)+frac{2t}{sin t}ln 2pi$
82. $intlimits_{0}^{+infty }{frac{{{left( sinh left( sx ight) ight)}^{2}}}{x{{left( {{e}^{x}}-1 ight)}^{3}}}dx}=log left( frac{2pi s}{sin left( 2pi s ight)} ight)quad 0<s<frac{1}{2}$
83. $intlimits_{0}^{+infty }{frac{{{x}^{s-1}}sinh left( pi x ight)}{{{left( cosh left( pi x ight)-1 ight)}^{3}}}dx}=frac{Gamma left( s ight)}{3{{pi }^{s}}}cdot left( zeta left( 4-s ight)-zeta left( 2-s ight) ight)$
84. $intlimits_{0}^{1}{intlimits_{0}^{1}{intlimits_{0}^{1}{sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}dxdydz}}}=log left( sqrt{3}+1 ight)-frac{log 2}{2}+frac{sqrt{3}}{4}-frac{pi }{24}$
85. $intlimits_{0}^{1}{intlimits_{0}^{1}{frac{{{x}^{alpha -1}}{{y}^{eta -1}}}{left( 1+xy ight)log left( xy ight)}dxdy}}=frac{1}{alpha -eta }cdot log left( frac{Gamma left( frac{alpha }{2} ight)Gamma left( frac{1}{2}+frac{eta }{2} ight)}{Gamma left( frac{eta }{2} ight)Gamma left( frac{1}{2}+frac{alpha }{2} ight)} ight)$
86. $intlimits_{-infty }^{+infty }{frac{1}{1+frac{{{x}^{2}}}{{{alpha }^{2}}}}cdot prodlimits_{k=1}^{+infty }{frac{1+frac{{{x}^{2}}}{{{left( eta +k ight)}^{2}}}}{1+frac{{{x}^{2}}}{{{left( alpha +k ight)}^{2}}}}dx}}=sqrt{pi }cdot frac{Gamma left( eta +1 ight)}{Gamma left( alpha ight)}cdot frac{Gamma left( alpha +frac{1}{2} ight)}{Gamma left( eta +frac{1}{2} ight)}cdot frac{Gamma left( eta -alpha +frac{1}{2} ight)}{Gamma left( eta -alpha +1 ight)}quad 0<alpha <eta +frac{1}{2}$
87. $intlimits_{0}^{+infty }{frac{1}{x}left( frac{sinh left( ax ight)}{sinh x}-a{{e}^{-2x}} ight)dx}=log left( frac{pi }{cos left( frac{api }{2} ight){{Gamma }^{2}}left( frac{a+1}{2} ight)} ight)$
88. $intlimits_{0}^{+infty }{{{x}^{2}}{{e}^{-{{x}^{2}}}}erfleft( x ight)log xdx}=frac{2-log 2}{16}sqrt{pi }-frac{gamma +log 2}{16sqrt{pi }}left( pi +2 ight)+frac{G}{4sqrt{pi }}$
89. $intlimits_{0}^{frac{pi }{2}}{sin left( 2nx ight)sinh left( asin x ight)sin left( acos x ight)dx}={{left( -1 ight)}^{n+1}}frac{pi }{4}cdot frac{{{a}^{2n}}}{left( 2n ight)!}$
90. Let $eta >0$ and $alpha in left( -frac{pi }{2},frac{pi }{2} ight)$. Prove that $intlimits_{0}^{+infty }{{{e}^{-tcos alpha }}{{t}^{eta -1}}cos left( tsin alpha ight)dx}=Gamma left( eta ight)cos left( eta sin alpha ight)$
91. $intlimits_{0}^{+infty }{frac{ln left( 1+x ight)ln left( 1+frac{1}{{{x}^{2}}} ight)}{x}dx}=pi G-frac{3}{8}zeta left( 3 ight)$
92. $intlimits_{-infty }^{+infty }{intlimits_{-infty }^{+infty }{ ext{sign}left( x ight) ext{sign}left( y ight){{e}^{-frac{{{x}^{2}}+{{y}^{2}}}{2}}}sin left( xy ight)dxdy}=2sqrt{2}log left( 1+sqrt{2} ight)}$
93. $intlimits_{0}^{+infty }{left( frac{x}{{{log }^{2}}left( {{e}^{{{x}^{2}}}}-1 ight)}-frac{x}{sqrt{{{e}^{{{x}^{2}}}}-1}{{log }^{2}}left( {{e}^{{{x}^{2}}}}-1 ight)}-frac{x}{sqrt{{{e}^{{{x}^{2}}}}-1}log left( {{left( {{e}^{{{x}^{2}}}}-1 ight)}^{2}} ight)} ight)dx}=frac{G}{pi }$
94. $intlimits_{0}^{1}{{{B}_{2n+1}}left( x ight)cot left( pi x ight)dx}=frac{2left( 2n+1 ight)!}{{{left( -1 ight)}^{n+1}}{{left( 2pi ight)}^{2n+1}}}zeta left( 2n+1 ight)$ where ${{B}_{2n+1}}left( x ight)$ is the Bernoulli Polynomial
95. $intlimits_{0}^{+infty }{frac{x}{1+{{x}^{4}}}arctan left( frac{psin qx}{1+pcos qx} ight)dx}=frac{pi }{2}arctan left( frac{psin left( frac{q}{sqrt{2}} ight)}{{{e}^{frac{q}{sqrt{2}}}}+pcos left( frac{q}{sqrt{2}} ight)} ight)$
96. Let $min Re$ and $ain left( -1,1
ight)$ Calculate
$intlimits_{0}^{2pi }{frac{{{e}^{mcos heta }}left( cos left( msin heta
ight)-asin left( heta +msin heta
ight)
ight)}{1-2asin heta +{{a}^{2}}}d heta }$
97. Prove that $intlimits_{0}^{1}{intlimits_{0}^{1}{frac{{{left( xy ight)}^{s-1}}{{y}^{n}}}{left( 1-xy ight)log left( xy ight)}dxdy}}=frac{Gamma 'left( s ight)}{Gamma left( s ight)}-frac{log left( n! ight)}{n}$
98. $intlimits_{0}^{+infty }{sin left( nx ight)left( cot x+coth x ight){{e}^{-nx}}dx}=frac{pi }{2}cdot frac{sinh left( npi ight)}{cosh left( npi ight)-cos left( npi ight)}$
99. $intlimits_{0}^{frac{pi }{3}}{{{log }^{2}}left( frac{sin x}{sin left( x+frac{pi }{3} ight)} ight)dx}=frac{5{{pi }^{3}}}{81}$
100. $intlimits_{0}^{2pi }{{{x}^{2}}log left( 1-exp left( ix ight) ight)dx}=2pi zeta left( 4 ight)-8i{{pi }^{2}}zeta left( 3 ight)$
100+1. Let $ain left( 0,1 ight)$ Prove that $intlimits_{0}^{1}{frac{log log frac{1}{x}}{1+2xcos left( api ight)+{{x}^{2}}}dx}=frac{pi }{2sin left( api ight)}left( alog left( 2pi ight)+log frac{Gamma left( frac{1}{2}+frac{a}{2} ight)}{Gamma left( frac{1}{2}-frac{a}{2} ight)} ight)$
Bonus
1. $intlimits_{0}^{+infty }{frac{cos x}{x}{{left( intlimits_{0}^{x}{frac{sin t}{t}dt}
ight)}^{2}}dx}=-frac{7}{6}zeta left( 3
ight)$
2. $intlimits_{0}^{+infty }{frac{{{x}^{a-1}}sin x}{cos x+cosh x}dx}={{2}^{1-frac{a}{2}}}Gamma left( a ight)sin left( frac{api }{4} ight)left( 1-{{2}^{1-a}} ight)zeta left( a ight)$
3. $intlimits_{0}^{+infty }{frac{cos left( tx ight)}{left( 1+{{x}^{2}} ight)cosh left( frac{pi x}{2} ight)}dx}=cosh tlog left( 2cosh t ight)-tsinh t$
4. $intlimits_{0}^{+infty }{frac{{{t}^{s-1}}}{{{z}^{-1}}{{e}^{t}}-1}dt}=Gamma left( s ight) ext{L}{{ ext{i}}_{s}}left( z ight)$
5. $intlimits_{0}^{+infty }{exp left( -2u ight)left( frac{1}{usinh u}-frac{1}{{{u}^{2}}coshu} ight)du}=2-log 2-frac{4G}{pi }$
6. $intlimits_{0}^{+infty }{frac{sin left( bt ight)}{t}{{e}^{-at}}ln tdt}=-left( gamma +frac{ln left( {{a}^{2}}+{{b}^{2}} ight)}{2} ight)arctan frac{b}{a}$
7. $intlimits_{0}^{1}{frac{left( a-t ight)ln left( 1-t ight)}{1-2at+{{t}^{2}}}dt}=frac{{{pi }^{2}}}{12}-frac{{{left( arccos a-pi ight)}^{2}}}{8}-frac{{{ln }^{2}}left( 2-2a ight)}{8}$
8. $intlimits_{0}^{1}{{{left{ frac{1}{x} ight}}^{2}}dx}=ln left( 2pi ight)-1-gamma$
9. $intlimits_{0}^{1}{{{left{ frac{1}{x} ight}}^{2}}left{ frac{1}{1-x} ight}dx}=2+gamma -ln left( 4pi ight)$
10. $intlimits_{0}^{1}{{{left{ frac{x}{y} ight}}^{2}}dxdy}=frac{1}{2}ln left( 2pi ight)-frac{1}{3}-frac{gamma }{2}$