1.∫(f(x)+g(x))dx=∫f(x)dx+∫g(x)dx2.∫(f(x)−g(x))dx=∫f(x)dx−∫g(x)dx3.∫f(x)dg(x)=f(x)g(x)−∫g(x)df(x)4.∫a x dx=a xln a+C，a≠1，a>05.∫x n dx=x n+1n+1+C，n≠−16.∫1xdx=ln|x|+C7.∫e x dx=e x+C8.∫sin x dx=−cos x+C9.∫cos x dx=sin x+C10.∫sec2x dx=tan x+C11.∫csc2x dx=−cot x+C12.∫sec x tan x dx=sec x+C13.∫csc x cot x dx=−csc x+C14.∫(ax+b)n dx=(ax+b)n+1a(n+1)+C，a≠0，n≠−115.∫dxax+b =1aln|ax+b|+C，a≠016.∫x(ax+b)n dx=(ax+b)n+1a2(ax+bn+2−bn+1)+C，a≠0，n≠−1,−217.∫xax+b dx=xa−ba2ln|ax+b|+C，a≠018.∫x(ax+b)2dx=1a2(ln|ax+b|+bax+b)+C，a≠019.∫x2ax+b dx=12a3[(ax+b)2−4b(ax+b)+2b2ln|ax+b|]+C20.∫x2(ax+b)2dx=1a3(ax+b−2b ln|ax+b|−b2ax+b)+C21.∫x2(ax+b)3dx=1a3(ln|ax+b|+2bax+b−b22(ax+b)2)+C22.∫x2(ax+b)n dx=1a3(−1(n−3)(ax+b)n−3+2b(n−2)(ax+b)n−2−b2(n−1)(ax+b)n−1)+C，n≠1,2,323.∫dxx(ax+b)=1bln|xax+b|+C，b≠024.∫dxx2(ax+b)=−1bx+ab2ln|ax+bx|+C25.∫dxx2(ax+b)2=−a(1b2(ax+b)+1ab2x−2b3ln|ax+bx|)+C26.∫x√ax+bdx=215a2(3ax−2b)(ax+b)32+C27.∫x2√ax+bdx=2105a3(15a2x2−12abx+8b2)(ax+b)32+C28.∫(√ax+b)n dx=2(√ax+b)n+2a(n+2)+C，a≠0，n≠−229.∫x n√ax+b dx=2a(2n+3)x n(ax+b)32−2nba(2n+3)∫x n−1√ax+bdx循环计算30.∫√ax+bx dx=2√ax+b+bx√ax+b=2√ax+b−2√b arctanh√ax+bb+C31.x√ax+b =√−b√ax+b−b+C，b<032.x√ax+b =√b|√ax+b−√b√ax+b+√b|+C，b>033.∫√ax+bx2dx=−√ax+bx+a2x√ax+b+C34.∫√ax+bx n dx=−(ax+b)32b(n−1)x n−1−(2n−5)a2b(n−1)∫√ax+bx n−1dx，n≠1循环计算35.n√ax+b =2a(2n+1)(x n√ax+b−bn n−1√ax+b)+C循环计算36.x2√ax+b =−ax+bbx−a2b x√ax+b+C，b≠037.x n√ax+b =−√ax+bb(n−1)x n−1−(2n−3)a2b(n−1)∫√ax+bx n−1dx，n≠1循环计算38.∫x n√ax+bdx=22n+1(x n+1√ax+b+bx n√ax+b−nb∫x n−1√ax+bdx)+C循环计算39. ∫dx a 2+x 2=1a arctan xa +C ，a ≠040. ∫dx(a 2+x 2)2=x2a 2(a 2+x 2)+12a 3arctan xa +C ，a ≠041. ∫dxa 2−x 2=12a ln |a+xa−x |+C =1a arctanh xa +C ，a ≠0，|a |>|x | 42. ∫dx(a 2−x 2)2=x2a 2(a 2−x 2)+14a 3ln |x+ax−a |+C43. ∫1x 2−a 2dx =12a ln |x−ax+a |+C =−1a arccoth xa +C ，a ≠0，|x |>|a | 44. √a 2+x 2=ln(x +√a 2+x 2)+C45. ∫√a 2+x 2dx =x2√a 2+x 2+a 22ln(x +√a 2+x 2)+C46. ∫(√a 2+x 2)3dx =x(√a 2+x 2)34+38a 2x√a 2+x 2+38a 4ln(x +√a 2+x 2)+C 47. ∫(√a 2+x 2)5dx =x(√a 2+x 2)56+524a2x(√a 2+x 2)3+516a 4x√a 2+x 2+516a 6ln(x +√a 2+x 2)+C48. ∫x(√a 2+x 2)2n+1dx =(√a 2+x 2)2n+32n+3+C 49. ∫x2√a 2+x 2dx=x8(a 2+2x2)√a 2+x 2−a 48ln(x +√a 2+x 2)+C50. ∫x 2(√a 2+x 2)3dx =x(√a 2+x 2)56−a 2x √a 2+x 224−a 4x √a 2+x 216−a 616ln(x +√a 2+x 2)+C 51. ∫x3√a 2+x 2dx=(√a 2+x 2)55−a 2(√a 2+x 2)33+C52. ∫x 3(√a 2+x 2)3dx =(√a 2+x 2)77−a 2(√a 2+x 2)55+C53. ∫x 3(√a 2+x 2)2n+1dx =(√a 2+x 2)2n+52n+5−a 2(√a 2+x 2)2n+32n+3+C54. ∫x4√a 2+x 2dx =x 3(√a 2+x 2)36−a 2x(√a 2+x 2)38+a 4x √a 2+x 216+a 616ln(x +√a 2+x 2)+C 55. ∫x 4(√a 2+x 2)3dx =x 3(√a 2+x 2)58−a 2x(√a 2+x 2)516+a 4x(√a 2+x 2)364+3a 6x √a 2+x 2128+3a 8128ln(x +√a 2+x 2)+C 56. ∫x 5√a 2+x 2dx =(√a 2+x 2)77−2a 2(√a 2+x 2)55+a 4(√a 2+x 2)33+C57. ∫x 5(√a 2+x 2)3dx =(√a 2+x 2)99−2a 2(√a 2+x 2)77+a 4(√a 2+x 2)55+C58. ∫x 5(√a 2+x 2)2n+1dx =(√a 2+x 2)2n+72n+7−2a 2(√a 2+x 2)2n+52n+5+a 4(√a 2+x 2)2n+32n+3+C59. ∫√a 2+x 2xdx =√a 2+x 2−a ln |a+√a 2+x 2x|+C =√a 2+x 2−a arcsinh ax +C 60. ∫(√a 2+x 2)3x dx =(√a 2+x 2)33+a2√a 2+x 2−a 3ln |a+√a 2+x 2x|+C61. ∫(√a 2+x 2)5x dx =(√a 2+x 2)55+a 2(√a 2+x 2)33+a 4√a 2+x 2−a 5ln |a+√a 2+x 2x|+C62. ∫(√a 2+x 2)77dx =(√a 2+x 2)77+a 2(√a 2+x 2)55+a 4(√a 2+x 2)33+a 6√a 2+x 2−a 7ln |a+√a 2+x 2x|+C63. ∫√a 2+x 2x 2dx =ln(x +√a 2+x 2)−√a 2+x 2x+C64. 2√a 2+x 2=−a 22ln(x +√a 2+x 2)+x √a 2+x 22+C =−a 22arcsinh xa +x √a 2+x 22+C65. x √a 2+x 2=−1a ln |a+√a 2+x 2x|+C =−1a arcsinh ax +C66. x 2√a 2+x 2=−√a 2+x 2a 2x +C ，a ≠067. √a 2−x 2=arcsin xa +C ，a ≠0，|x |≤|a |68. ∫√a 2−x 2dx =x 2√a 2−x 2+a 22arcsin xa +C ，a ≠0，|x |≤|a |69. ∫√a 2−x 2dx =12(x√a 2−x 2−sgn x arccosh |xa |)+C ，|x |≥|a | 70. ∫x√a 2−x 2dx =−(√a 2−x 2)33+C ，|x |≤|a |71. ∫x 2√a 2−x 2dx =a 48arcsin xa −18x√a 2−x 2(a 2−2x 2)+C ，a ≠0 72. ∫√a 2−x 2x dx =√a 2−x 2−a ln |a+√a 2−x 2x|+C ，|x |≤|a |73. ∫√a 2−x 2x 2dx =−arcsin xa −√a 2−x 2x +C ，a ≠074. 2√a 2−x 2=a 22arcsin xa −x √a 2−x 22+C ，a ≠0，x√a 2−x 275. x √a 2−x 2=−1a ln |a+√a 2−x 2x|+C ，a ≠076. x 2√a 2−x 2=−√a 2−x 2a 2x+C ，a ≠077. √x 2−a 2=ln|x +√x 2−a 2|+C 78. ∫√x 2−a 2dx =x2√x 2−a 2−a 22ln|x +√x 2−a 2|+C79. ∫(√x 2−a 2)ndx =x(√x 2−a 2)nn+1−na 2n+1∫(√x 2−a 2)n−2dx ，n ≠−1 循环计算 80. (√x 2−a 2)n=x(√x 2−a 2)2−n(2−n )a 2+n−3(2−n )a 2∫dx (√x 2−a 2)n−2，n ≠2 循环计算81. ∫x(√x 2−a 2)ndx =(√x 2−a 2)n+2n+2+C ，n ≠−2 82. ∫x 2√x 2−a 2dx =x8(2x 2−a2)√x 2−a 2−a 48ln|x +√x 2−a 2|+C83. ∫√x 2−a 2xdx =√x 2−a 2−a arcsec |xa |+C =√x 2−a 2−a arccos |ax |+C ，a ≠0 84. √x 2−a 2=√x 2−a 2+C 85. ∫x dx (√x 2−a 2)3=√x 2−a 2+C 86. ∫x dx (√x 2−a 2)5=−13(√x 2−a 2)3+C 87. ∫x dx (√x 2−a 2)7=−15(√x 2−a 2)5+C88. ∫x dx (√x 2−a 2)2n+1=−1(2n−1)(√x 2−a 2)2n−1+C89. ∫√x 2−a 2x 2dx =ln|x +√x 2−a 2|−√x 2−a 2x +C90. 2√x 2−a 2=a 22ln|x +√x 2−a 2|+x2√x 2−a 2+C91. ∫x 2(√x 2−a 2)3dx =√x 2−a2ln |x+√x 2−a 2a|+C 92. 4√x 2−a 2=x 3√x 2−a 24+38a 2x√x 2−a 2+38a 4ln |x+√x 2−a 2a|+C93. ∫x 4(√x 2−a 2)3dx =x √x 2−a 22−2√x 2−a 2+32a 2ln |x+√x 2−a 2a |+C 94. ∫x 4(√x 2−a 2)5dx =√x 2−a2x 33(√x 2−a 2)3+ln |x+√x 2−a 2a |+C95. ∫x 2m dx (√x 2−a 2)2n+1=−x 2m−1(2n−1)(√x 2−a 2)2n−1+2m−12n−1∫x 2m−2(√x 2−a 2)2n−1dx +C =(−1)n−ma 2(n−m )∑12(m+i )+1(n−m−1i )x 2(m+i )+1(√x 2−a 2)2(m+i )+1n−m−1i=0，n >m ≥096. ∫dx (√x 2−a 2)3=a 2√x 2−a 2+C 97. ∫dx (√x 2−a 2)5=1a 4(√x 2−a 2−x 33(√x 2−a 2)3)+C98. ∫dx (√x 2−a 2)7=−1a 6(√x 2−a 2−2x 33(√x 2−a 2)3+x 55(√x 2−a 2)5)+C99. ∫dx (√x 2−a 2)9=1a 8(√x 2−a 2−2x 33(√x 2−a 2)3+3x 55(√x 2−a 2)5−x 77(√x 2−a 2)7)+C100. ∫x 2(√x 2−a 2)5dx =−x 33a 2(√x 2−a 2)3+C101.∫x 2(√x 2−a 2)7dx =1a4(x 33(√x 2−a 2)3−x 55(√x 2−a 2)5)+C102. ∫x 2(√x 2−a 2)9dx =−1a 6(x 33(√x 2−a 2)3−2x 55(√x 2−a 2)5+x 77(√x 2−a 2)7)+C103. x √x 2−a 2=1a arcsec |xa |+C ，a ≠0 104. x 2√x 2−a 2=√x 2−a 2a 2x +C ，a ≠0105. ∫dx ax 2+bx+c =√4ac−b 2√4ac−b 24ac −b 2>0106.∫dxax 2+bx+c =√b 2−4ac √b 2−4ac =√b 2−4ac |√b 2−4ac2ax+b+√b 2−4ac |，4ac −b 2<0 107. ∫dxax 2+bx+c =−22ax+b ，4ac −b 2=0108. ∫dxax 2+bx+c =12a ln |ax 2+bx +c |−b2a ∫dxax 2+bx+c +C109.∫mx+nax 2+bx+c dx =m2a ln |ax 2+bx +c |+a √4ac−b 2√4ac−b 2+C ，4ac −b 2>0 110.∫mx+nax 2+bx+c dx =m2a ln |ax 2+bx +c |+a √b 2−4ac √b 2−4ac +C，4ac−b2<0111.∫mx+nax2+bx+c dx=m2aln|ax2+bx+c|−2an−bma(2ax+b)+C，4ac−b2=0112.∫dx(ax2+bx+c)n =2ax+b(n−1)(4ac−b2)(ax2+bx+c)n−1+(2n−3)2a(n−1)(4ac−b2)∫dx(ax2+bx+c)n−1+C113.∫x(ax2+bx+c)n dx=bx+2c(n−1)(4ac−b2)(ax2+bx+c)n−1−b(2n−3) (n−1)(4ac−b2)∫dx(ax2+bx+c)n−1+C114.∫dxx(ax2+bx+c)=12cln|x2ax2+bx+c|−b2c∫dxax2+bx+c+C115.√ax2+bx+c =√aln|2√a2x2+abx+ac+2ax+b|+C，a>0116.√ax2+bx+c =√a√4ac−b2+C，a>0，4ac−b2>0117.√ax2+bx+c =√a|2ax+b|+C，a>0，4ac−b2=0118.√ax2+bx+c =√−a√b2−4ac+C，a<0，4ac−b2<0119.∫dx(√ax2+bx+c)3=(4ac−b2)√ax2+bx+c+C120.∫dx(√ax2+bx+c)5=3(4ac−b2)√ax2+bx+c(1ax2+bx+c+8a4ac−b2)+C121.∫dx(√ax2+bx+c)2n+1=4ax+2b(2n−1)(4ac−b2)(√ax2+bx+c)2n−1+8a(n−1) (2n−1)(4ac−b2)∫dx(√ax2+bx+c)2n−1+C循环计算122.√ax2+bx+c =√ax2+bx+ca−b2a√ax2+bx+c+C123.∫x dx(√ax2+bx+c)3=(4ac−b2)√ax2+bx+c+C124.∫x dx(√ax2+bx+c)2n+1=−1(2n−1)a(√ax2+bx+c)2n−1−b2a∫dx(√ax2+bx+c)2n−1+C125.x√ax2+bx+c =√c(2√acx2+bcx+c2+bx+2cx)+C126.x√ax2+bx+c =√c(|x|√4ac−b2)+C127.∫sin2x dx=x2−sin2x4+C128.∫√1−sin x dx=∫√cvs x dx=2cos x2+sin x2cos x2−sin x2，√cvs x=2√1+sin x，其中cvsx是conversine函数129.∫sin n ax dx=−sin n−1ax cos axan +n−1n∫sin n−2ax dx+C循环计算130.∫sin axx dx=∑(−1)i(ax)2i+1(2i+1)(2i+1)!∞i=0+C131.∫sin axx n dx=−sin ax(n−1)x n−1+an−1∫cos axx n−1dx132.∫cos n ax dx=1an cos n−1ax sin ax+n−1n∫cos n−2ax dx+C，n≥2133.∫cos2x dx=x2+sin2x4+C134.∫cos axx dx=ln|ax|+∑(−1)i(ax)2i2i(2i)!∞i−1，n≠1135.∫cos axx n dx=−cos ax(n−1)x n−1−an−1∫sin axx n−1dx，n≠1136.∫sin ax cos ax dx=12asin2ax137.∫sin ax sin bx dx=sin[(a−b)x]2(a−b)−sin[(a+b)x]2(a+b)+C，a2≠b2138.∫sin ax cos bx dx=−cos[(a+b)x]2(a+b)−cos[(a−b)x]2(a−b)+C，a2≠b2139.∫cos ax cos bx dx=sin[(a−b)x]2(a−b)+sin[(a+b)x]2(a+b)+C，a2≠b2140.∫sin ax cos ax dx=−cos2ax4a+C，a≠0141.∫sin n ax cos ax dx=sin n+1ax(n+1)a+C，a≠0，n≠−1142.∫cos n ax sin ax dx=−cos n+1ax(n+1)a+C，a≠0，n≠−1143.∫tan ax dx=∫sin axcos ax dx=−1aln|cos ax|+C，a≠0144.∫cot ax dx=∫cos axsin ax dx=1aln|sin ax|+C，a≠0145.∫sin n ax cos m ax dx=−sin n−1ax cos m+1axa(m+n)+n−1 m+n ∫sin n−2ax cos m ax dx+C=sin n+1ax cos m−1axa(m+n)+m−1n+m∫sin n ax cos m−2ax dx+C，a≠0，m+n≠0循环计算146.∫sin ax sin bx dx=x sin(a−b)2(a−b)−x sin(a+b)2(a+b)+C，|a|≠|b|147.∫dxsin ax cos ax =1aln|tan ax|+C148.∫dxsin ax cos n ax =1a(n−1)cos n−1ax+∫dxsin ax cos n−2ax，n≠1149.∫dxcos ax sin n ax =−1a(n−1)sin n−1ax+∫dxcos ax sin n−2ax，n≠1150.∫sin axdxcos n ax =1a(n−1)cos n−1ax+C，n≠1151.∫sin2axdxcos ax =−1asin ax+1aln|tan(π4+ax2)|+C152.∫sin2axdxcos n ax =sin axa(n−1)cos n−1ax−1n−1∫dxcos n−2ax，n≠1153.∫sin n axdxcos ax =−sin n−1axa(n−1)+∫sin n−2axdxcos ax+C154.∫sin n axdxcos m ax =sin n+1axa(m−1)cos m−1ax−n−m+2m−1∫sin n axdxcos m−2ax+C=−sin n−1axa(n−m)cos m−1ax +n−1n−m∫sin n−2axdxcos m ax+C=sin n−1axa(m−1)cos m−1ax−n−1 m−1∫sin n−1axdxcos m−2ax+C，m≠1，m≠n155.∫cos axdxsin n ax =−1a(n−1)sin n−1ax+C156.∫cos2axdxsin ax =1a(cos ax+ln|tan ax2|)+C157.∫cos2axdxsin n ax =−1n−1(cos axa sin n−1ax+∫dxsin n−2ax)+C，n≠1158.∫cos n axdxsin m ax =−cos n+1axa(m−1)sin m−1ax−n−m−2m−1∫cos n axdxsin m−2ax+C=cos n−1axa(n−m)sin m−1ax+n−1 n−m ∫cos n−2axdxsin m ax+C=−cos n−1axa(m−1)sin m−1ax−n−1m−1∫cos n−2axdxsin m−2ax+C，m≠1，m≠n159.∫dxb+c sin ax =a√b2−c2|√b−cb+ctan(π4−ax2)|+C，a≠0，b2>c2160.∫dxb+c sin ax =a√c2−b2|c+b sin ax+√c2−b2cos axb+c sin ax|+C，a≠0，b2<c2161.∫dx1+sin ax =−1atan(π4−ax2)+C，a≠0162.∫dx1−sin ax =1atan(π4+ax2)+C，a≠0163.∫x dx1+sin ax =xatan(ax2−π4)+2c2ln|cos(ax2−π4)|+C164.∫x dx1−sin ax =xacot(π4−ax2)+2c2ln|sin(π4−ax2)|+C165.∫sin axdx1±sin ax =±x+1ctan(π4∓ax2)+C166.∫dxb+c cos ax =a√b2−c2|√b−cb+ctan ax2|+C，a≠0，b2>c2167.∫dxb+c cos ax =a√c2−b2|c+b cos ax+√c2−b2sin axb+c cos ax|+C，a≠0，b2<c2168.∫dx1+cos ax =1atan ax2+C，a≠0169.∫dx1−cos ax =−1acot ax2+C，a≠0170.∫x dx1+cos ax =xatan ax2+2a2ln|cos ax2|+C，a≠0171.∫x dx1−cos ax =−xacot ax2+2a2ln|sin ax2|+C，a≠0172.∫cos axdx1+cos ax =x−1atan ax2+C173.∫cos axdx1−cos ax =−x−1acot ax2+C174.∫cos ax cos bx dx=x sin(a−b)2(a−b)+x sin(a+b)2(a+b)+C，|a|≠|b|175.∫dxcos ax±sin ax =√2a|tan(ax2±π8)|+C176.∫dx(cos ax+sin ax)2=12atan(ax∓π4)+C177.∫dx(cos x+sin x)n =1n−1(sin x−cos x(cos x+sin x)n−1−2(n−2)∫dx(cos x+sin x)n−2)+C178.∫dx(cos ax+sin ax)n=179.∫cos axdxcos ax+sin ax =x2+12aln|sin ax+cos ax|+C180.∫cos axdxcos ax−sin ax =x2−12aln|sin ax−cos ax|+C181.∫sin axdxcos ax+sin ax =x2−12aln|sin ax+cos ax|+C182.∫sin axdxcos ax−sin ax =x2−12aln|sin ax−cos ax|+C183.∫cos axdxsin ax(1+cos ax)=−14atan2ax2+12aln|tan ax2|+C184.∫cos axdxsin ax(1−cos ax)=−14acot2ax2−12aln|tan ax2|+C185.∫sin axdxcos ax(1+sin ax)=14acot2(ax2+π4)+12aln|tan(ax2+π4)|+C186.∫sin axdxcos ax(1−sin ax)=14atan2(ax2+π4)−12aln|tan(ax2+π4)|+C187.∫sin ax tan ax dx=1a(ln|sec ax+tan ax|−sin ax)+C188.∫tan n axdxsin2ax =1a(n−1)tan n−1ax，n≠1189.∫tan n axdxcos2ax =1a(n+1)tan n+1ax，n≠−1190.∫cot n axdxsin2ax =1a(n+1)cot n+1ax，n≠−1191.∫cot n axdxcos2ax =1a(1−n)tan1−n ax，n≠1192.∫tan m axcot n ax =1a(m+n−1)tan m+n−1ax−∫tan m−2axcot n axdx，m+n≠1193.∫x sin ax dx=1a2sin ax−xacos ax+C，a≠0194.∫x cos ax dx=cos axa2+x sin axa+C195.∫x n sin ax dx=−x na cos ax+na∫x n−1cos ax dx，a≠0循环计算196.∫x n cos ax dx=x na sin ax−na∫x n−1sin ax dx，a≠0循环计算197.∫tan ax dx=−1aln|cos ax|+C，a≠0198.∫cot ax dx=1aln|sin ax|+C，a≠0199.∫tan2ax dx=1atan ax−x+C，a≠0200.∫cot2ax dx=−1acot ax−x+C，a≠0201.∫tan n ax dx=tan n−1axa(n−1)−∫tan n−2ax dx，a≠0，n≠1循环计算202.∫cot n ax dx=−cot n−1axa(n−1)−∫cot n−2ax dx，a≠0，n≠1循环计算203.∫dxtan ax+1=x2+12aln|sin ax+cos ax|+C204.∫dxtan ax−1=−x2+12aln|sin ax−cos ax|+C205.∫tan axdxtan ax+1=x2−12aln|sin ax+cos ax|+C206.∫tan axdxtan ax−1=x2+12aln|sin ax−cos ax|+C207.∫dx1+cot ax =∫tan axdxtan ax+1=x2−12aln|sin ax+cos ax|+C208.∫dx1−cot ax =∫tan axdxtan ax−1=x2+12aln|sin ax−cos ax|+C209.∫sec ax dx=1a ln|sec ax+tan ax|+C=1aln|tan(ax2+π4)|，a≠0210.∫csc ax dx=−1a ln|csc ax+cot ax|+C=1aln|tan ax2|+C，a≠0211.∫sec n ax dx=sec n−2ax tan axa(n−1)+n−2n−1∫sec n−2ax dx，a≠0，n≠1循环计算212.∫csc n ax dx=−csc n−2ax cot axa(n−1)+n−2n−1∫csc n−2ax dx，a≠0，n≠1循环计算213.∫sec n ax tan ax dx=sec n axna+C，a≠0，n≠0214.∫csc n ax cot ax dx=−csc n axna+C，a≠0，n≠0215.∫dxsec x+1=x−tan x2+C216.∫arcsin ax dx=x arcsin ax+1a√1−a2x2+C，a≠0217.∫x arcsin xa dx=(x22−a24)arcsin xa+x4√c2−x2+C218.∫x2arcsin xa dx=x33arcsin xa+x2+2c29√c2−x2+C219.∫x n arcsin x dx=1n+1(x n+1arcsin x+x n√1−x2−nx n−1arcsin xn−1+n∫x n−2arcsin x dx)+C220.∫arccos ax dx=x arccos ax−1a√1−a2x2+C，a≠0221.∫x arccos xa dx=(x22−a24)arccos xa−x4√a2−x2+C222.∫x2arccos xa dx=x33arccos xa−x2+2a29√a2−x2+C223.∫arctan ax dx=x arctan ax−12aln(1+a2x2)+C，a≠0224.∫x arctan xa dx=a2+x22arctan xa−ax2+C225.∫x2arctan xa dx=x33arctan xa−ax26+a36ln a2+x2+C226.∫x n arctan xa dx=x n+1n+1arctan xa−an+1∫x n+1a2+x2dx+C，n≠1227.∫arccot ax dx=x arccot ax+12aln(1+a2x2)+C228.∫x arccot xa dx=a2+x22arccot xa+ax2+C229.∫x2arccot xa dx=x33arccot xa+ax26−a36ln(a2+x2)+C230.∫x n arccot xa dx=x n+1n+1arccot xa+an+1∫x n+1a2+x2dx，n≠1231.∫arcsec ax dx=x arcsec ax+ax|x|ln(x±√x2−1)+C232.∫x arcsec x dx=12(x2arcsec x−√x2−1)+C233.∫x n arcsec x dx=1n+1(x n+1arcsec x−1n[x n−1√x2−1+(1−n)(x n−1arcsec x+(1−n)∫x n−2arcsec x dx)])+C234.∫arccsc ax dx=x arccsc ax−ax|x|ln(x±√x2−1)+C235.∫sinh ax dx=1acosh ax+C236.∫cosh ax dx=1asinh ax+C237.∫sinh2ax dx=14a sinh2ax−x2+C238.∫cosh2ax dx=14a sinh2ax+x2+C239.∫sinh n ax dx=1an sinh n−1ax cosh ax−n−1n∫sinh n−2ax dx+C，n>0 =1a(n+1)sinh n+1ax cosh ax−n+2n+1∫sinh n+2ax dx+C，n<0,n≠−1240.∫cosh n ax dx=1an sinh ax cosh n−1ax+n−1n∫cosh n+2ax dx，n<0,n≠−1241.∫dxsinh ax =1aln|tanh ax2|+C=1aln|cosh ax−1sinh ax|+C=1aln|sinh axcosh ax+1|+C=1 a ln|cosh ax−1cosh ax+1|+C242.∫dxcosh ax =2aarctan e ax+C243.∫dxsinh n ax =cosh axa(n−1)sinh n−1ax−n−2n−1∫dxsinh n−2ax，n≠1244.∫dxcosh n ax =sinh axa(n−1)cosh n−1ax+n−2n−1∫dxcosh n−2ax，n≠1245.∫cosh n axsinh m ax dx=cosh n−1axa(n−m)sinh m−1ax+n−1n−m∫cosh n−2axsinh m axdx=−cosh n+1axa(m−1)sinh m−1ax +n−m+2m−1∫cosh n axsinh m−2axdx+C=−cosh n−1axa(m−1)sinh m−1ax+n−1 m−1∫cosh n−2axsinh m−2axdx+C，m≠n,m≠1246.∫sinh m axcos n ax dx=sinh m−1axa(m−n)cosh n−1ax+m−1m−n∫sinh m−2axcosh n axdx+C=sinh m+1axa(n−1)cosh n−1ax +m−n+2n−1∫sinh m axcosh n−2axdx+C=sinh m−1axa(n−1)cosh n−1ax+m−1 n−1∫sinh m−2axcosh n−2axdx+C，m≠n,n≠1247.∫x sinh ax dx=1a x cosh ax−1a2sinh ax+C248.∫x cosh ax dx=1a x sinh ax−1a2cosh ax+C249.∫tanh ax dx=1aln|cosh ax|+C250.∫coth ax dx=1aln|sinh ax|+C251.∫tanh n ax dx=−tanh n−1axa(n−1)+∫tanh n−2ax dx+C，n≠1252.∫coth n ax dx=−coth n−1axa(n−1)+∫coth n−2ax dx，n≠1253.∫sinh ax sinh bx dx=a sinh bx cosh ax−b cosh bx sinh axa2−b2+C254.∫cosh ax cos bx dx=a sinh ax cosh bx−b sinh bx cosh axa2−b2+C255.∫cosh ax sinh bx dx=a sinh ax sinh bx−b cosh ax cosh bxa2−b2+C256.∫sinh(ax+b)sin(cx+d)dx=aa2+c2cosh(ax+b)sin(cx+d)−ca2+c2sinh(ax+b)cos(cx+d)+C257.∫sinh(ax+b)cos(cx+d)dx=aa2+c2cosh(ax+b)cos(cx+d)+ca2+c2sinh(ax+b)sin(cx+d)+C258.∫cosh(ax+b)sin(cx+d)dx=aa2+c2sinh(ax+b)sin(cx+d)−ca2+c2cosh(ax+b)cos(cx+d)+C259.∫cosh(ax+b)cos(cx+d)dx=aa2+c2sinh(ax+b)cos(cx+d)+ca2+c2cosh(ax+b)sin(cx+d)+C260.∫arcsinh xa dx=x arcsinh xa−√x2+a2+C261.∫arccosh xa dx=x arccosh xa−√x2−a2+C262.∫arctanh xa dx=x arctanh xa+a2ln|a2−x2|+C，|x|<|a|263.∫arccoth xa dx=x arccoth xa+a2ln|x2−a2|+C，|x|<|a|264.∫arcsech xa dx=x arcsech xa−a arctanx√a−xa+xx−a+C，x∈(0,a)265.∫arccsch xa dx=x arccsch xa+a ln x+√x2+a2a+C，x∈(0,a)266.∫xe ax dx=e axa2(ax−1)+C，a≠0267.∫b ax dx=b axa lnb+C，a≠0，b>0，b≠1268.∫x2e ax dx=e ax(x2a −2xa2+2a3)+C269.∫x n e ax dx=x n e axa −na∫x n−1e ax dx，a≠0270.∫e ax dxx =ln|x|+∑(ax)ii·i!∞i=1+C271.∫e ax dxx n =1n−1(−e axx n−1+a∫e axx n−1dx)+C，n≠1272.∫e ax ln x dx=1ae ax ln|x|−Ei(ax)+C273.∫e ax sin bx dx=e axa2+b2(a sin bx−b cos bx)+C274. ∫e axcos bx dx =e axa 2+b 2(a cos bx +b sin bx )+C 275. ∫e ax sin nbx dx =e ax sin n−1x a 2+n 2(a sin x −n cos x )+n (n−1)a 2+n 2∫e ax sin n−2x dx276.∫eaxcos n bx dx =e ax cos n−1xa 2+n 2(a cos x +n sin x )+n (n−1)a 2+n 2∫e ax cos n−2x dx277. ∫xe ax 2dx =12a e ax 2+C 278. σ√2π−(x−μ)22σ2dx =12σ(1+√2σ+C279.∫e x 2dx =ex 2(∑a 2jx 2j+1n−1j=0)+(2n −1)a 2n−2∫e x2x 2n dx ，n >0 其中a 2j =1·3·5···(2j−1)2j+1=2j!j!22j+1+C280. ∫e −ax 2dx ∞−∞=√πa the Gaussian integral281. ∫x 2n e −x 2a 2dx∞0=√π(2n )!n!(a 2)2n+1282. ∫ln ax dx =x ln ax −x +C283. ∫(ln x )2dx =x (ln x )2−2x ln x +2x +C 284. ∫(ln ax )n dx =x (ln ax )n −n ∫(ln ax )n−1dx 285. ∫dxln x=ln |ln x |+ln x +∑(ln x )i i·i!∞i=2+C286. ∫dx (ln x )n =−x(n−1)(ln x )n−1+1n−1∫dx(ln x )n−1+C ，n ≠1 287. ∫x m ln x dx =x m+1(ln xm+1−1(m+1)2)+C ，n ≠−1 288. ∫x m (ln x )n dx =x m+1(ln x )nm+1−nm+1∫x m (ln x )n−1dx +C ，m ≠−1289. ∫(ln x )n dxx =(ln x )n+1n+1+C ，n ≠−1290. ∫ln xdx x m =−ln x(m−1)x m−1−1(m−1)2x m−1，m ≠1 291. ∫(ln x )n dxx m=−(ln x )n(m−1)x m−1+nm−1∫(ln x )n−1dxx m，m ≠1 292. ∫x m dx (ln x )n=−x m+1(n−1)(ln x )n−1+m+1n−1∫x m dx(ln x )n−1，n ≠1293.∫x n (ln ax)mdx =x n+1(ln ax )mn+1−mn+1∫x n (ln ax )m−1dx ，n ≠−1294.∫(ln ax)mx dx=(ln ax)m+1m+1+C，m≠−1295.∫dxx ln ax=ln|ln ax|+C296.∫dxx n ln x =ln|ln x|+∑(−1)i(n−1)i(ln x)ii·i!∞i=1+C297.∫dxx(ln x)n =−1(n−1)(ln x)n−1，n≠1298.∫sin(ln x)dx=x2[sin(ln x)−cos(ln x)]+C299.∫cos(ln x)dx=x2[sin(ln x)+cos(ln x)]+C300.∫e x(x ln x−x−1x)dx=e x(x ln x−x−ln x)+C。