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)dx=1a(ax+b−2b ln|ax+b|−b2ax+b)+C21.∫x2(ax+b)dx=1a(ln|ax+b|+2bax+b−b22(ax+b))+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 −a dx =12a ln |x−ax+a |+C =−1a arccoth xa +C ，a ≠0，|x |>|a | 44. 22=ln(x +√a 2+x 2)+C45. ∫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+C57.58. 22=ln|x +√x 2−a 2|+C 59. ∫√x 2−a 2dx =x2√x 2−a 2−a 22ln|x +√x 2−a 2|+C60. ∫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 61. (√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62. ∫x(√x 2−a 2)ndx =(√x 2−a 2)n+2n+2+C ，n ≠−2 63. ∫x 2√x 2−a 2dx =x8(2x 2−a2)√x 2−a 2−a 48ln|x +√x 2−a 2|+C64. ∫√x 2−a 2xdx =√x 2−a 2−a arcsec |xa |+C =√x 2−a 2−a arccos |ax |+C ，a ≠0 65. √22=√x 2−a 2+C 66. ∫x dx (√x 2−a 2)3=√22+C 67. ∫x dx (√x 2−a 2)5=−13(√x 2−a 2)3+C 68. ∫x dx (√x 2−a 2)7=−15(√x 2−a 2)5+C69. ∫x dx (√x 2−a 2)2n+1=−1(2n−1)(√x 2−a 2)2n−1+C70. ∫√x 2−a 2x 2dx =ln|x +√x 2−a 2|−√x 2−a 2x +C71. 2√22=a 22ln|x +√x 2−a 2|+x2√x 2−a 2+C72. ∫x 2(√x 2−a 2)3dx =√22ln |x+√x 2−a 2a|+C 73. 4√22=x 3√x 2−a 24+38a 2x√x 2−a 2+38a 4ln |x+√x 2−a 2a|+C74. ∫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 75. ∫x 4(√x 2−a 2)5dx =√x 2−a2x 33(√x 2−a 2)3+ln |x+√x 2−a 2a |+C76. ∫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 ≥077. ∫dx (√x 2−a 2)3=222+C 78. ∫dx (√x 2−a 2)5=1a (√22−x 33(√x 2−a 2)3)+C79. ∫dx (√x 2−a 2)7=−1a (√22−2x 33(√x 2−a 2)3+x 55(√x 2−a 2)5)+C80. ∫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)+C81. ∫x 2(√x 2−a 2)5dx =−x 33a 2(√x 2−a 2)3+C82. ∫x 2(√x 2−a 2)7dx =1a4(x 33(√x 2−a 2)3−x 55(√x 2−a 2)5)+C83. ∫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)+C84. √22=1a arcsec |xa |+C ，a ≠0 85. 2√22=√x 2−a 2a 2x +C ，a ≠0 86. ∫dxax +bx+c =√2√24ac −b 2>0 87. ∫dxax 2+bx+c =√b 2−4ac√b 2−4ac=√b 2−4ac|√b 2−4ac 2ax+b+√b 2−4ac|，4ac −b 2<088. ∫dx ax +bx+c =−22ax+b ，4ac −b 2=089. ∫dxax 2+bx+c =12a ln |ax 2+bx +c |−b2a ∫dxax 2+bx+c +C 90. ∫mx+nax 2+bx+c dx =m2a ln |ax 2+bx +c |+√2√2+C ，4ac −b 2>091. ∫mx+nax 2+bx+c dx =m2a ln |ax 2+bx +c |+√2√2+C ，4ac −b2<092.∫mx+nax2+bx+c dx=m2aln|ax2+bx+c|−2an−bma(2ax+b)+C，4ac−b2=093.∫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+C94.∫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+C95.∫dxx(ax2+bx+c)=12cln|x2ax2+bx+c|−b2c∫dxax2+bx+c+C96.√2=√aln|2√a2x2+abx+ac+2ax+b|+C，a>097.√ax2+bx+c =√a√4ac−b2+C，a>0，4ac−b2>098.√ax2+bx+c =√a|2ax+b|+C，a>0，4ac−b2=099.√2=−a√2+C，a<0，4ac−b2<0100.∫dx(√ax2+bx+c)3=(2)√2+C101.∫dx(√ax2+bx+c)5=(2)√2(1ax2+bx+c+8a4ac−b2)+C102.∫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103.√2=√ax2+bx+ca−b2a√2+C104.∫x dx(√ax2+bx+c)3=(2)√2+C105.∫x dx(√ax2+bx+c)2n+1=−1(2n−1)a(√ax2+bx+c)2n−1−b2a∫dx(√ax2+bx+c)2n−1+C106.√2=√c(2√acx2+bcx+c2+bx+2cx)+C107.x√ax2+bx+c =√c(|x|√4ac−b2)+C108.∫sin2x dx=x2−sin2x4+C109.∫√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110.∫sin n ax dx=−sin n−1ax cos axan +n−1n∫sin n−2ax dx+C循环计算111.∫sin axx dx=∑(−1)i(ax)2i+1(2i+1)(2i+1)!∞i=0+C112.∫sin axx dx=−sin ax(n−1)x+an−1∫cos axxdx113.∫cos n ax dx=1an cos n−1ax sin ax+n−1n∫cos n−2ax dx+C，n≥2114.∫cos2x dx=x2+sin2x4+C115.∫cos axx dx=ln|ax|+∑(−1)i(ax)2i2i(2i)!∞i−1，n≠1116.∫cos axx n dx=−cos ax(n−1)x n−1−an−1∫sin axx n−1dx，n≠1117.∫sin ax cos ax dx=12asin2ax118.∫sin ax sin bx dx=sin[(a−b)x]2(a−b)−sin[(a+b)x]2(a+b)+C，a2≠b2119.∫sin ax cos bx dx=−cos[(a+b)x]2(a+b)−cos[(a−b)x]2(a−b)+C，a2≠b2120.∫cos ax cos bx dx=sin[(a−b)x]2(a−b)+sin[(a+b)x]2(a+b)+C，a2≠b2121.∫sin ax cos ax dx=−cos2ax4a+C，a≠0122.∫sin n ax cos ax dx=sin n+1ax(n+1)a+C，a≠0，n≠−1123.∫cos n ax sin ax dx=−cos n+1ax(n+1)a+C，a≠0，n≠−1124.∫tan ax dx=∫sin axcos ax dx=−1aln|cos ax|+C，a≠0125.∫cot ax dx=∫cos axsin ax dx=1aln|sin ax|+C，a≠0126.∫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127.∫sin ax sin bx dx=x sin(a−b)2(a−b)−x sin(a+b)2(a+b)+C，|a|≠|b|128.∫dxsin ax cos ax =1aln|tan ax|+C129.∫dxsin ax cos n ax =1a(n−1)cos n−1ax+∫dxsin ax cos n−2ax，n≠1130.∫dxcos ax sin ax =−1a(n−1)sin ax+∫dxcos ax sin ax，n≠1131.∫sin axdxcos ax =1a(n−1)cos ax+C，n≠1132.∫sin2axdxcos ax =−1asin ax+1aln|tan(π4+ax2)|+C133.∫sin2axdxcos n ax =sin axa(n−1)cos n−1ax−1n−1∫dxcos n−2ax，n≠1134.∫sin n axdxcos ax =−sin n−1axa(n−1)+∫sin n−2axdxcos ax+C135.∫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 ax+C，m≠1，m≠n136.∫cos axdxsin n ax =−1a(n−1)sin n−1ax+C137.∫cos2axdxsin ax =1a(cos ax+ln|tan ax2|)+C138.∫cos2axdxsin n ax =−1n−1(cos axa sin n−1ax+∫dxsin n−2ax)+C，n≠1139.∫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 ax+C=−cos n−1axa(m−1)sin ax−n−1m−1∫cos n−2axdxsin ax+C，m≠1，m≠n140.∫dxb+c sin ax =a√b2−c2|√b−cb+ctan(π4−ax2)|+C，a≠0，b2>c2141.∫dxb+c sin ax =√22|c+b sin ax+√c2−b2cos axb+c sin ax|+C，a≠0，b2<c2142.∫dx1+sin ax =−1atan(π4−ax2)+C，a≠0143.∫dx1−sin ax =1atan(π4+ax2)+C，a≠0144.∫x dx1+sin ax =xatan(ax2−π4)+2c2ln|cos(ax2−π4)|+C145.∫x dx1−sin ax =xacot(π4−ax2)+2c2ln|sin(π4−ax2)|+C146.∫sin axdx1±sin ax =±x+1ctan(π4∓ax2)+C147.∫dxb+c cos ax =√22|√b−cb+ctan ax2|+C，a≠0，b2>c2148.∫dxb+c cos ax =√22|c+b cos ax+√c2−b2sin axb+c cos ax|+C，a≠0，b2<c2149.∫dx1+cos ax =1atan ax2+C，a≠0150.∫dx1−cos ax =−1acot ax2+C，a≠0151.∫x dx1+cos ax =xatan ax2+2aln|cos ax2|+C，a≠0152.∫x dx1−cos ax =−xacot ax2+2a2ln|sin ax2|+C，a≠0153.∫cos axdx1+cos ax =x−1atan ax2+C154.∫cos axdx1−cos ax =−x−1acot ax2+C155.∫cos ax cos bx dx=x sin(a−b)2(a−b)+x sin(a+b)2(a+b)+C，|a|≠|b|156.∫dxcos ax±sin ax =√2a|tan(ax2±π8)|+C157.∫dx(cos ax+sin ax)2=12atan(ax∓π4)+C158.∫dx(cos x+sin x)=1n−1(sin x−cos x(cos x+sin x)−2(n−2)∫dx(cos x+sin x))+C159.∫dx(cos ax+sin ax)n=160.∫cos axdxcos ax+sin ax =x2+12aln|sin ax+cos ax|+C161.∫cos axdxcos ax−sin ax =x2−12aln|sin ax−cos ax|+C162.∫sin axdxcos ax+sin ax =x2−12aln|sin ax+cos ax|+C163.∫sin axdxcos ax−sin ax =x2−12aln|sin ax−cos ax|+C164.∫cos axdxsin ax(1+cos ax)=−14atan2ax2+12aln|tan ax2|+C165.∫cos axdxsin ax(1−cos ax)=−14acot2ax2−12aln|tan ax2|+C166.∫sin axdxcos ax(1+sin ax)=14acot2(ax2+π4)+12aln|tan(ax2+π4)|+C167.∫sin axdxcos ax(1−sin ax)=14atan2(ax2+π4)−12aln|tan(ax2+π4)|+C168.∫sin ax tan ax dx=1a(ln|sec ax+tan ax|−sin ax)+C169.∫tan n axdxsin2ax =1a(n−1)tan n−1ax，n≠1170.∫tan n axdxcos2ax =1a(n+1)tan n+1ax，n≠−1171.∫cot n axdxsin2ax =1a(n+1)cot n+1ax，n≠−1172.∫cot n axdxcos ax =1a(1−n)tan1−n ax，n≠1173.∫tan m axcot n ax =1a(m+n−1)tan m+n−1ax−∫tan m−2axcot n axdx，m+n≠1174.∫x sin ax dx=1a2sin ax−xacos ax+C，a≠0175.∫x cos ax dx=cos axa +x sin axa+C176.∫x n sin ax dx=−x na cos ax+na∫x n−1cos ax dx，a≠0循环计算177.∫x n cos ax dx=x na sin ax−na∫x n−1sin ax dx，a≠0循环计算178.∫tan ax dx=−1aln|cos ax|+C，a≠0179.∫cot ax dx=1aln|sin ax|+C，a≠0180.∫tan2ax dx=1atan ax−x+C，a≠0181.∫cot2ax dx=−1acot ax−x+C，a≠0182.∫tan n ax dx=tan n−1axa(n−1)−∫tan n−2ax dx，a≠0，n≠1循环计算183.∫cot n ax dx=−cot n−1axa(n−1)−∫cot n−2ax dx，a≠0，n≠1循环计算184.∫dxtan ax+1=x2+12aln|sin ax+cos ax|+C185.∫dxtan ax−1=−x2+12aln|sin ax−cos ax|+C186.∫tan axdxtan ax+1=x2−12aln|sin ax+cos ax|+C187.∫tan axdxtan ax−1=x2+12aln|sin ax−cos ax|+C188.∫dx1+cot ax =∫tan axdxtan ax+1=x2−12aln|sin ax+cos ax|+C189.∫dx1−cot ax =∫tan axdxtan ax−1=x2+12aln|sin ax−cos ax|+C190.∫sec ax dx=1a ln|sec ax+tan ax|+C=1aln|tan(ax2+π4)|，a≠0191.∫csc ax dx=−1a ln|csc ax+cot ax|+C=1aln|tan ax2|+C，a≠0192.∫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193.∫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194.∫sec n ax tan ax dx=sec n axna+C，a≠0，n≠0195.∫csc n ax cot ax dx=−csc n axna+C，a≠0，n≠0196.∫dxsec x+1=x−tan x2+C197.∫arcsin ax dx=x arcsin ax+1a√1−a2x2+C，a≠0198.∫x arcsin xa dx=(x22−a24)arcsin xa+x4√c2−x2+C199.∫x2arcsin xa dx=x33arcsin xa+x2+2c29√c2−x2+C200.∫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)+C201.∫arccos ax dx=x arccos ax−1a√1−a2x2+C，a≠0202.∫x arccos xa dx=(x22−a24)arccos xa−x4√a2−x2+C203.∫x2arccos xa dx=x33arccos xa−x2+2a29√a2−x2+C204.∫arctan ax dx=x arctan ax−12aln(1+a2x2)+C，a≠0205.∫x arctan xa dx=a2+x22arctan xa−ax2+C206.∫x2arctan xa dx=x33arctan xa−ax26+a36ln a2+x2+C207.∫x n arctan xa dx=x n+1n+1arctan xa−an+1∫x n+1a+xdx+C，n≠1208.∫arccot ax dx=x arccot ax+12aln(1+a2x2)+C209.∫x arccot xa dx=a2+x22arccot xa+ax2+C210.∫x2arccot xa dx=x33arccot xa+ax26−a36ln(a2+x2)+C211.∫x n arccot xa dx=x n+1n+1arccot xa+an+1∫x n+1a2+x2dx，n≠1212.∫arcsec ax dx=x arcsec ax+ax|x|ln(x±√x2−1)+C213.∫x arcsec x dx=12(x2arcsec x−√x2−1)+C214.∫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)])+C215.∫arccsc ax dx=x arccsc ax−ax|x|ln(x±√x2−1)+C216.∫sinh ax dx=1acosh ax+C217.∫cosh ax dx=1asinh ax+C218.∫sinh2ax dx=14a sinh2ax−x2+C219.∫cosh2ax dx=14a sinh2ax+x2+C220.∫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≠−1221.∫cosh n ax dx=1an sinh ax cosh n−1ax+n−1n∫cosh n+2ax dx，n<0,n≠−1222.∫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|+C223.∫dxcosh ax =2aarctan e ax+C224.∫dxsinh n ax =cosh axa(n−1)sinh n−1ax−n−2n−1∫dxsinh n−2ax，n≠1225.∫dxcosh ax =sinh axa(n−1)cosh ax+n−2n−1∫dxcosh ax，n≠1226.∫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≠1227.∫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 ax +m−n+2n−1∫sinh m axcosh axdx+C=sinh m−1axa(n−1)cosh ax+m−1 n−1∫sinh m−2axcosh axdx+C，m≠n,n≠1228.∫x sinh ax dx=1a x cosh ax−1a2sinh ax+C229.∫x cosh ax dx=1a x sinh ax−1acosh ax+C230.∫tanh ax dx=1aln|cosh ax|+C231.∫coth ax dx=1aln|sinh ax|+C232.∫tanh n ax dx=−tanh n−1axa(n−1)+∫tanh n−2ax dx+C，n≠1233.∫coth n ax dx=−coth n−1axa(n−1)+∫coth n−2ax dx，n≠1234.∫sinh ax sinh bx dx=a sinh bx cosh ax−b cosh bx sinh axa2−b2+C235.∫cosh ax cos bx dx=a sinh ax cosh bx−b sinh bx cosh axa2−b2+C236.∫cosh ax sinh bx dx=a sinh ax sinh bx−b cosh ax cosh bxa2−b2+C237.∫sinh(ax+b)sin(cx+d)dx=aa2+c2cosh(ax+b)sin(cx+d)−ca2+c2sinh(ax+b)cos(cx+d)+C238.∫sinh(ax+b)cos(cx+d)dx=aa2+c2cosh(ax+b)cos(cx+d)+ca2+c2sinh(ax+b)sin(cx+d)+C239.∫cosh(ax+b)sin(cx+d)dx=aa+csinh(ax+b)sin(cx+d)−ca+ccosh(ax+b)cos(cx+d)+C240.∫cosh(ax+b)cos(cx+d)dx=aa2+c2sinh(ax+b)cos(cx+d)+ca2+c2cosh(ax+b)sin(cx+d)+C241.∫arcsinh xa dx=x arcsinh xa−√x2+a2+C242.∫arccosh xa dx=x arccosh xa−√x2−a2+C243.∫arctanh xa dx=x arctanh xa+a2ln|a2−x2|+C，|x|<|a|244.∫arccoth xa dx=x arccoth xa+a2ln|x2−a2|+C，|x|<|a|245.∫arcsech xa dx=x arcsech xa−a arctanx√a−xa+xx−a+C，x∈(0,a)246.∫arccsch xa dx=x arccsch xa+a ln x+√x2+a2a+C，x∈(0,a)247.∫xe ax dx=e axa2(ax−1)+C，a≠0248.∫b ax dx=b axa lnb+C，a≠0，b>0，b≠1249.∫x2e ax dx=e ax(x2a −2xa2+2a3)+C250.∫x n e ax dx=x n e axa −na∫x n−1e ax dx，a≠0251.∫e ax dxx =ln|x|+∑(ax)ii·i!∞i=1+C252.∫e ax dxx n =1n−1(−e axx n−1+a∫e axx n−1dx)+C，n≠1253.∫e ax ln x dx=1ae ax ln|x|−Ei(ax)+C254.∫e ax sin bx dx=e axa+b(a sin bx−b cos bx)+C255.∫e ax cos bx dx=e axa2+b2(a cos bx+b sin bx)+C256. ∫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 dx257.∫eaxcos nbx 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 dx258. ∫xe ax 2dx =12a e ax 2+C 259. σ√2π−(x−μ)22σ2dx =12σ(1+√2σ+C260.∫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+C261. ∫e −ax 2dx ∞−∞=√πa the Gaussian integral262. ∫x 2n e −x 2a 2dx∞0=√π(2n )!n!(a 2)2n+1263. ∫ln ax dx =x ln ax −x +C264. ∫(ln x )2dx =x (ln x )2−2x ln x +2x +C 265. ∫(ln ax )n dx =x (ln ax )n −n ∫(ln ax )n−1dx 266. ∫dxln x =ln |ln x |+ln x +∑(ln x )ii·i!∞i=2+C267. ∫dx(ln x )=−x(n−1)(ln x )+1n−1∫dx(ln x )+C ，n ≠1 268. ∫x m ln x dx =x m+1(ln xm+1−1(m+1)2)+C ，n ≠−1 269. ∫x m (ln x )ndx =x m+1(ln x )nm+1−nm+1∫x m (ln x )n−1dx +C ，m ≠−1270. ∫(ln x )n dxx =(ln x )n+1n+1+C ，n ≠−1 271. ∫ln xdx x m =−ln x (m−1)x m−1−1(m−1)2x m−1，m≠1272. ∫(ln x )n dxx m=−(ln x )n(m−1)x m−1+nm−1∫(ln x )n−1dxx m，m ≠1 273. ∫x m dx (ln x )=−x m+1(n−1)(ln x )+m+1n−1∫x m dx(ln x )，n ≠1274. ∫x n (ln ax )mdx =x n+1(ln ax )mn+1−mn+1∫x n (ln ax )m−1dx ，n ≠−1275.∫(ln ax )mxdx =(ln ax )m+1m+1+C ，m ≠−1276.∫dxx ln ax=ln|ln ax|+C277.∫dxx n ln x =ln|ln x|+∑(−1)i(n−1)i(ln x)ii·i!∞i=1+C278.∫dxx(ln x)=−1(n−1)(ln x)，n≠1279.∫sin(ln x)dx=x2[sin(ln x)−cos(ln x)]+C280.∫cos(ln x)dx=x2[sin(ln x)+cos(ln x)]+C281.∫e x(x ln x−x−1x)dx=e x(x ln x−x−ln x)+C。