常用积分公式 (一)含有ar+b的积分(a≠0) 1jaa-na+4+c 2.J(x+bydk-(x+by+C(uz-1) a(u+1) .(nkxC jn-Ba+-2a+bnC .c at=la+a产4c .(kC dx (二)含有V√arx+b的积分 n小+t=名m+可+C 1.∫+6d=1品(r-2bax++C 2jr+=0aor-12+wnm+6+( 2 6t-是a-2n+5+c
常 用 积 分 公 式 (一)含有 ax b 的积分( a 0 ) 1. dx ax b = 1 ln ax b C a 2. ( ) d ax b x = 1 1 ( ) ( 1) ax b C a ( 1 ) 3. d x x ax b = 2 1 ( ln ) ax b b ax b C a 4. 2 d x x ax b = 2 2 3 1 1 ( ) 2 ( ) ln 2 ax b b ax b b ax b C a 5. d ( ) x x ax b = 1 ln ax b C b x 6. 2 d ( ) x x ax b = 2 1 ln a ax b C bx b x 7. 2 d ( ) x x ax b = 2 1 (ln ) b ax b C a ax b 8. 2 2 d ( ) x x ax b = 2 3 1 ( 2 ln ) b ax b b ax b C a ax b 9. 2 d ( ) x x ax b = 2 1 1 ln ( ) ax b C b ax b b x (二)含有 ax b 的积分 10. ax b x d = 2 3 ( ) 3 ax b C a 11. x ax b x d = 3 2 2 (3 2 ) ( ) 15 ax b ax b C a 12. 2 x ax b x d = 2 2 2 3 3 2 (15 12 8 ) ( ) 105 a x abx b ax b C a 13. d x x ax b = 2 2 ( 2 ) 3 ax b ax b C a
√ar+b-√b +C(b>0) dx ax+b+/b 2 arctan ax+ 2+C(b0)的积分 arctan x+C (b>0) 22 人 r √ab x2+b 1n ax-F-b 2√-ab"Wax+√-b +C(b<0) 23. a+=++c
14. 2 d x x ax b = 2 2 2 3 2 (3 4 8 ) 15 a x abx b ax b C a 15. dx x ax b = 1 ln ( 0) 2 arctan ( 0) ax b b C b b ax b b ax b C b b b 16. 2 dx x ax b = d 2 ax b a x bx b x ax b 17. d ax b x x = d 2 x ax b b x ax b 18. 2 d ax b x x = d 2 ax b a x x x ax b (三)含有 2 2 x a 的积分 19. 2 2 dx x a = 1 arctan x C a a 20. 2 2 d ( )n x x a = 2 2 2 1 2 2 2 1 2 3 d 2( 1) ( ) 2( 1) ( ) n n x n x n a x a n a x a 21. 2 2 dx x a = 1 ln 2 x a C a x a (四)含有 2 ax b a ( 0) 的积分 22. 2 dx ax b = 1 arctan ( 0) 1 ln ( 0) 2 a x C b ab b ax b C b ab ax b 23. 2 d x x ax b = 1 2 ln 2 ax b C a
24. 25. a+ dx 1. x2 2++C -In dx 26. ∫x+ =-1_af_dx bx b ax?+b 27. dx h xa+b) ax'+b1 +C dx 28. 1 dx (ar2+b)2 2b(ax2+b)2b ax2+b (五)含有ax2+bx+c(a>0)的积分 2 2ax+b arctan +C (b24ac) b2-4ac 2ax+b+b2-4ac dx (六)含有Vx2+a2(a>0)的积分 z2-a后+G-+Fa+C dx ac dx a=++C 34
24. 2 2 d x x ax b = 2 x b xd a a ax b 25. 2 d ( ) x x ax b = 2 2 1 ln 2 x C b ax b 26. 2 2 d ( ) x x ax b = 2 1 d a x bx b ax b 27. 3 2 d ( ) x x ax b = 2 2 2 2 1 ln 2 2 a ax b C b x bx 28. 2 2 d ( ) x ax b = 2 2 1 d 2 ( ) 2 x x b ax b b ax b (五)含有 2 ax bx c ( 0) a 的积分 29. 2 dx ax bx c = 2 2 2 2 2 2 2 2 2 arctan ( 4 ) 4 4 1 2 4 ln ( 4 ) 4 2 4 ax b C b ac ac b ac b ax b b ac C b ac b ac ax b b ac 30. 2 d x x ax bx c = 2 2 1 d ln 2 2 b x ax bx c a a ax bx c (六)含有 2 2 x a ( 0) a 的积分 31. 2 2 dx x a = 1 arsh x C a = 2 2 ln( ) x x a C 32. 2 2 3 d ( ) x x a = 2 2 2 x C a x a 33. 2 2 d x x x a = 2 2 x a C 34. 2 2 3 d ( ) x x x a = 2 2 1 C x a
2X1 36. oda+x+F++C + 37. -In a 38. -+C a'x jF+a-F++x+F+a)+C 0.∫G+ac=营2x+5iF+口+ax+F++c 4.xods-y+c 2.∫r+ad=营r+aF+-ginx+r+a)+c a.c (七)含有Vx2-a2(a>0)的积分 s高高c-a+-d+c dx .∫ad=-+C
35. 2 2 2 d x x x a = 2 2 2 2 2 ln( ) 2 2 x a x a x x a C 36. 2 2 2 3 d ( ) x x x a = 2 2 2 2 ln( ) x x x a C x a 37. 2 2 dx x x a = 2 2 1 ln x a a C a x 38. 2 2 2 dx x x a = 2 2 2 x a C a x 39. 2 2 x a x d = 2 2 2 2 2 ln( ) 2 2 x a x a x x a C 40. 2 2 3 ( ) d x a x = 2 2 2 2 4 2 2 3 (2 5 ) ln( ) 8 8 x x a x a a x x a C 41. 2 2 x x a x d = 1 2 2 3 ( ) 3 x a C 42. 2 2 2 x x a x d = 4 2 2 2 2 2 2 (2 ) ln( ) 8 8 x a x a x a x x a C 43. 2 2 d x a x x = 2 2 2 2 ln x a a x a a C x 44. 2 2 2 d x a x x = 2 2 2 2 ln( ) x a x x a C x (七)含有 2 2 x a ( 0) a 的积分 45. 2 2 dx x a = 1 arch x x C x a = 2 2 ln x x a C 46. 2 2 3 d ( ) x x a = 2 2 2 x C a x a 47. 2 2 d x x x a = 2 2 x a C
49. jn--a+号邮F-+c 50. 应可 51. *C -arccos 52. dx --d.c 5.小-adk=F-a-号nr+原-a+c s4.S-ya-x-sc 5.∫R-adr=}-arT+c s.-as-e--m+c 7.j=F--a+C gk-区r+原-可c (八)含有Va-x2(a>0)的积分 dx dx
48. 2 2 3 d ( ) x x x a = 2 2 1 C x a 49. 2 2 2 d x x x a = 2 2 2 2 2 ln 2 2 x a x a x x a C 50. 2 2 2 3 d ( ) x x x a = 2 2 2 2 ln x x x a C x a 51. 2 2 dx x x a = 1 arccos a C a x 52. 2 2 2 dx x x a = 2 2 2 x a C a x 53. 2 2 x a x d = 2 2 2 2 2 ln 2 2 x a x a x x a C 54. 2 2 3 ( ) d x a x = 2 2 2 2 4 2 2 3 (2 5 ) ln 8 8 x x a x a a x x a C 55. 2 2 x x a x d = 1 2 2 3 ( ) 3 x a C 56. 2 2 2 x x a x d = 4 2 2 2 2 2 2 (2 ) ln 8 8 x a x a x a x x a C 57. 2 2 d x a x x = 2 2 arccos a x a a C x 58. 2 2 2 d x a x x = 2 2 2 2 ln x a x x a C x (八)含有 2 2 a x ( 0) a 的积分 59. 2 2 dx a x =arcsin x C a 60. 2 2 3 d ( ) x a x = 2 2 2 x C a a x
6L∫后a-F-+c 63. 2 arcsin+C a 64. _dx Va-x :-arcsin+C a 65. a-va-x+C x dx 66. 6.∫F-xdr=F-平+gar arcsin'+C 2 a 8小后-ryt=d-2rn后-+esin2+C w.Sxl-xas--jla-+c 0.∫rG-d=2r-n-r+g arcsin 小k-F7+aha-G-C x a (九)含有√±r2+bx+c(a>0)的积分 d i-LIn2ax+b+2ayax+bx+c+C 73.ar+bx+c val
61. 2 2 d x x a x = 2 2 a x C 62. 2 2 3 d ( ) x x a x = 2 2 1 C a x 63. 2 2 2 d x x a x = 2 2 2 arcsin 2 2 x a x a x C a 64. 2 2 2 3 d ( ) x x a x = 2 2 arcsin x x C a x a 65. 2 2 dx x a x = 2 2 1 ln a a x C a x 66. 2 2 2 dx x a x = 2 2 2 a x C a x 67. 2 2 a x x d = 2 2 2 arcsin 2 2 x a x a x C a 68. 2 2 3 ( ) d a x x = 2 2 2 2 4 3 (5 2 ) arcsin 8 8 x x a x a x a C a 69. 2 2 x a x x d = 1 2 2 3 ( ) 3 a x C 70. 2 2 2 x a x x d = 4 2 2 2 2 (2 ) arcsin 8 8 x a x x a a x C a 71. 2 2 d a x x x = 2 2 2 2 ln a a x a x a C x 72. 2 2 2 d a x x x = 2 2 arcsin a x x C x a (九)含有 2 ax bx c ( 0) a 的积分 73. 2 dx ax bx c = 1 2 ln 2 2 ax b a ax bx c C a
74.∫小Nm2+bx+cdr=2ar+b ax+bx+c Aa +4a-1l2x+b+、ax+b 8v园 xdx=+bx+c 75.∫a+bx+c a 6a+++网 dx .2ax-b_+C √b2+4ac n小+-md=26ra-a+“ .2ax-b_+C -arcsin- 8a b2+4ac 2ax-b -arcsin- +C Q B2+4ac +)含有,或-a6-可的银分 9.∫-g=--g+-an-4+k-刚+c [x-a+C Vb-x dx e小-oo-w-29-a6-,m4c (a<b) (十一)含有三角函数的积分 83.sin xdx =-cosx+C
74. 2 ax bx c x d = 2 2 4 ax b ax bx c a 2 2 3 4 ln 2 2 8 ac b ax b a ax bx c C a 75. 2 d x x ax bx c = 1 2 ax bx c a 2 3 ln 2 2 2 b ax b a ax bx c C a 76. 2 dx c bx ax = 2 1 2 arcsin 4 ax b C a b ac 77. 2 c bx ax x d = 2 2 3 2 2 4 2 arcsin 4 8 4 ax b b ac ax b c bx ax C a a b ac 78. 2 d x x c bx ax = 2 3 2 1 2 arcsin 2 4 b ax b c bx ax C a a b ac (十)含有 x a x b 或 ( )( ) x a b x 的积分 79. d x a x x b =( ) ( )ln( ) x a x b b a x a x b C x b 80. d x a x b x =( ) ( )arcsin x a x a x b b a C b x b x 81. d ( )( ) x x a b x =2arcsin x a C b x ( ) a b 82. ( )( )d x a b x x = 2 2 ( ) ( )( ) arcsin 4 4 x a b b a x a x a b x C b x ( ) a b (十一)含有三角函数的积分 83. sin dx x = cos x C
84.「cosxdx=sinx+C 85.tan xdx=-In cosx+C 86. [cot xdx Insinx+C 87. ∫Sscxdx=-lnan经+c=Ineex+tam时+C 88. fesexdst 89.「sec2xdr=tanx+C 90. ∫csc2xdr=-cotx+C 91.sec xtanxdx=secx+C 92.[cscxcotxdx =-cscx+C 93.∫sin2xdr=x-1s sin2x+C 24 94.∫cos2dr=+ sin2x+C 24 95.fsinsinxcsinxd 1 96.[cos"xdx=-cos"-xsinx+ n l∫cos2xd n 97. sin"x n-1 sin"-x n-1J sin"-2x 98. sinx n-2 dx cos"x n-1 cos"-1 x n-1J cos"-2x 99. [cos"xsin"xdx=1cosxsin[cosxsin"xdx m+n m+nJ 、1 cosm xsinx1 cos"xsin"-2 xdx m+n m+n 1 l00.「sinax cos bxdx=- 1 cos(a+b)x- cos(a-b)x+C 2(a+b) 2(a-b)
84. cos dx x =sin x C 85. tan dx x = ln cos x C 86. cot dx x =ln sin x C 87. sec dx x =ln tan( ) 4 2 x C =ln sec tan x x C 88. csc dx x =ln tan 2 x C =ln csc cot x x C 89. 2 sec dx x = tan x C 90. 2 csc dx x = cot x C 91. sec tan d x x x =sec x C 92. csc cot d x x x = csc x C 93. 2 sin dx x = 1 sin 2 2 4 x x C 94. 2 cos dx x = 1 sin 2 2 4 x x C 95. sin d n x x = 1 1 1 2 sin cos sin d n n n x x x x n n 96. cos d n x x = 1 1 1 2 cos sin cos d n n n x x x x n n 97. d sinn x x = 1 2 1 cos 2 d 1 sin 1 sin n n x n x n x n x 98. d cosn x x = 1 2 1 sin 2 d 1 cos 1 cos n n x n x n x n x 99. cos sin d m n x x x = 1 1 1 1 2 cos sin cos sin d m n m n m x x x x x m n m n = 1 1 1 1 2 cos sin cos sin d m n m n n x x x x x m n m n 100. sin cos d ax bx x = 1 1 cos( ) cos( ) 2( ) 2( ) a b x a b x C a b a b
101.[sinaxsin bxdx 1 -sin(a+b)x+2(a-b) 1 -sin(a-b)x +C 102.cosax cos bxdx= 1一sina+br+2a-b° 2(a+b) 1,sin(a-b)x+C X dx 2 atan~+b 103. arctan 2 +C(a2>b2) a+bsinx a2-b2 Va-b2 dx 1 atan'+b-√6-a 104.」 In +C(a2b2) a+b a+b tan dx 1 a+bIn 106.J a+bcosx a+616-1 2 b-a +C(a20) 1l3.∫arcsin'dr=xaresin'+√2-x+C a a
101. sin sin d ax bx x = 1 1 sin( ) sin( ) 2( ) 2( ) a b x a b x C a b a b 102. cos cos d ax bx x = 1 1 sin( ) sin( ) 2( ) 2( ) a b x a b x C a b a b 103. d sin x a b x = 2 2 2 2 tan 2 2 arctan x a b C a b a b 2 2 ( ) a b 104. d sin x a b x = 2 2 2 2 2 2 tan 1 2 ln tan 2 x a b b a C x b a a b b a 2 2 ( ) a b 105. d cos x a b x = 2 arctan( tan ) 2 a b a b x C a b a b a b 2 2 ( ) a b 106. d cos x a b x = tan 1 2 ln tan 2 x a b a b b a C a b b a x a b b a 2 2 ( ) a b 107. 2 2 2 2 d cos sin x a x b x = 1 arctan( tan ) b x C ab a 108. 2 2 2 2 d cos sin x a x b x = 1 tan ln 2 tan b x a C ab b x a 109. x ax x sin d = 2 1 1 sin cos ax x ax C a a 110. 2 x ax x sin d = 2 2 3 1 2 2 x ax x ax ax C cos sin cos a a a 111. x ax x cos d = 2 1 1 cos sin ax x ax C a a 112. 2 x ax x cos d = 2 2 3 1 2 2 x ax x ax ax C sin cos sin a a a (十二)含有反三角函数的积分(其中 a 0 ) 113. arcsin dx x a = 2 2 arcsin x x a x C a
114. )aresin-x+C a 24 a 4 Il5.∫aresind=xar resin+x2+2a2后-r+C a 3 a 9 116.Jarccosdx=xarecos+C a 117. x2a2、 g)arccos-后-r+C a a 4 118. ∫arccos'dr=arcco arccos1 (x2+2a2)a-x2+C 3 119. [arctan dx xarctan-4In(a+x)+C a a 2 120. [xarctan dx =(a'+x)arctana -x+C a 2 a 2 21.∫arctan-号arcta言-gr+2lnMo+xr)+C a 3 a66 (十三)含有指数函数的积分 122.Sa'dx-La+C Ina 123.∫edr=1e"+c 1 a 124.∫xedr= a(ax-l)e"+C 12s.jret=dre-gJred jd-品+C 126 127.∫x"a'dr= Ina 128.∫e“sinbxdx= (asin b-bcosbx)+C 12.ood(n)
114. arcsin d x x x a = 2 2 2 2 ( )arcsin 2 4 4 x a x x a x C a 115. 2 arcsin dx x x a = 3 1 2 2 2 2 arcsin ( 2 ) 3 9 x x x a a x C a 116. arccos dx x a = 2 2 arccos x x a x C a 117. arccos dx x x a = 2 2 2 2 ( )arccos 2 4 4 x a x x a x C a 118. 2 arccos d x x x a = 3 1 2 2 2 2 arccos ( 2 ) 3 9 x x x a a x C a 119. arctan d x x a = 2 2 arctan ln( ) 2 x a x a x C a 120. arctan d x x x a = 1 2 2 ( )arctan 2 2 x a a x x C a 121. 2 arctan d x x x a = 3 3 2 2 2 arctan ln( ) 3 6 6 x x a a x a x C a (十三)含有指数函数的积分 122. d x a x = 1 ln x a C a 123. e dax x = 1 e ax C a 124. e dax x x = 2 1 ( 1)eax ax C a 125. e d n ax x x = 1 1 e e d n ax n ax n x x x a a 126. d x xa x = 2 1 ln (ln ) x x x a a C a a 127. d n x x a x = 1 1 d ln ln n x n x n x a x a x a a 128. e sin d ax bx x = 2 2 1 e ( sin cos ) ax a bx b bx C a b 129. e cos d ax bx x = 2 2 1 e ( sin cos ) ax b bx a bx C a b