12.（一）含有ax b 的积分（a 1. dx 1 ax b a =-In ax b 2. 3. 4. 5. 6. 7.9. 10.11. 13. 常用积分公式0)1 (ax b) dx = a( 1) x 1 dx = -^(ax b ax b a 丄dx =丄ax ba 3(ax bln b)2 b)ax b) C 2b(ax b) b 2lnax bdx x( ax b) dx x 2(ax b) x 2dx (ax b)2(^dx1ln b 1 bxax ax b1=-r(ln aax bax b )2bln ax b b 2 ax b ) Cdx 2x(ax b) b(ax b)含有.ax b 的积分12In b 2ax bTax~ dx = — T(ax~b)3 3a x 、、ax bdx = -^(3ax 2b15ax 2 . ax bdx = ^^(15a 2x 2 12abx 8b 2) ., (ax b)3 C105a).(ax b)3 Cx2- d x = -- 2 (ax 2b)、ax b C ，ax b 3a 2215a 3dx x ¥ ax b dx x 21 ax bax b. dx = (3a 2x 2 4abx 8b 2)、、ax b■, ax b 、.； b .ax b.b AC (b(b 0)0)bx 2b x 丫 ax b2 ax bdx x, ax bax b ,2 dx =xa dx 2 x 、ax b14.15.16.17.18.(三)19.20.21 .(四)22.23.含有x 2 a 2的积分dx 1 x arcta n—C22 _xa a adxx2n 3 dx (x2a 2) n= 「2(n 1)a 2/22、n 1(x a )2( n 1)a 222 n 1(x a )dx~22 =x a含有ax 2 b(a 0)的积分dx ax 2 b~^= arcta n j ax C Vbax b2TZ ln(b 0) C (b 0)x2ax -dx b2aax 2 -In 2a2 34.24. x 2—- ax b dx ax 2 b25. dx x(ax 2 b) In 2b 2x 2~ ax26. dx x 2(ax 2 b) 1 bx27.dx3a l 3 2 2lnx 3(ax 2 b)2b 228. dx 2 2(ax b)29.30. (六) 31. 32. 33. dx ~2- ax2ax b2x2b(ax 2 b) 12bdx ax 2 b含有ax 2bx dx ax 2 bx cx dx ax bx c 含有■. x 2dx dx.(x 2 a 2)x7x 2" c(a 0)的积分-=2 2 arctan —2ax b _ C ■. 4ac b 、4ac ba 2 (a 丄 ln ax 2a arsh- a—dx = 2 a1dx = ----- = C/ 2 2■ x a2 ax b J b 24ac 2 ax b J b 2 4ac C b .b 2 4ac(b 2 4ac)2(b 4ac)2 bx c 0)的积分 dx ___ 2a ax 2 bx c C 1 = ln(x 、.x 2 a 2) C ■. x 2 a 2 C22—x _____ dx =7^ d,(x a )ln( x \ x 2 a 2)dx 2 2 x x a2 2x a 」 dx =xdx "x^O 2xarch 凶 aG =ln xdxx :=a 2 dx =\ x 2 a 2dx35.36.37.38.39.40.41 .42.43.44.(七)45.46.47.dxx i x 2 a 2 lln aX '. x 2 a 2dx = 、 x dx = (2x8 1 J( 3 5a 2)、.x 2X 2 X 2 a2dx = 8(2x ln( x . x 2 a 2) C 4a ln(x 8、.x 2 a 2) C2 2x a 」2 dx = x x 2 a 2ln(x、x 2 a 2)含有\ x 2a 2 (a 0)的积分—22.x a2 a xa 2)■- x 2 a 2___ x a 2 x 2.(x 2 a 2)32 ~~2, x 2x x a dx =(2x 8 aa arccos —xIn x含有、•.a 2 x 2(a 0)的积分=arcs in 仝 Cadxxx 」1dx =CJ\3I 22a ) x a2 x 2 2 x a 2r a a ln2 2 x . dx =2 2、3 x a ) x . ln2 2x a1=—arccosx 、x * 2 a 2 a dx x 2adx x 2 x^? 2 2x a 2a xC In 2 2x x a48.49.50.51 .52.53.54. 55.56.57.58.(八)59.60.4—ln 8x2a x■- x 2 2 adx =x22a . x arcs in C 2 aarcsin - C a,(a /x 2)3dxdx x 、a 2 x 2 dx x 2、a 2 x 2 lln a 心 x 2 a2 2.a x 2 a x 2a . x arcsin3a 4arcs in 仝 C8~ x = x ~x 2X 2 Ldx =評X 2 / 2 2 ax, 2 2----------- dx = '. a x _2 2 ~2 2 a x a x 2 ----dx =—x2、 2 2a ) .a x4a . x arcs in 8 aa Ina a 2 x 2 QCxarcsin 一 Ca含有i ax 2 bx c (a 0)的积分 dx 1 2= = ln 2ax b Vaxbx c 7a61.62.63.64.65.66.67.68. 69.70.71 .72. (九) 73.= 2 ax b —2=cdx = ----------- ：,ax bx c4a（十一）含有三角函数的积分82.C (a b)74.ax bx75.x ax 2 bxdx =76.dx c bx ax 277.、、c bx ax2dx4ac b 2 8 . a 3In 2ax b 2石 bxc C1Jax 2bx c a— In 2ax b 2梟罷 2府1 2 ax b —i= arcsi n = C ■ a . b 4ac2ax b - ’ 2c bx ax 4ax 2 bx cb 2 4ac . 2ax barcs inC8 . a 3 b 2 4ac78..xdx = Sc bx ax 2—空 arcsin —2ax —b — C.c bx ax 2a2 . a 3b 2 4acJ x b) Cb)F (b 加® F C=2arcsi n_a C (a b) x a)(b x)1b x(x a)(b x)的积分b 后（b曲79.83. sin xdx = cosx Ccosxdx = sinx Ccot xdx = ln si nx C sec xdx = ln ta n(- -) C 4 2 csc xdx = ln tan — C =ln2sec xdx = tan x Ccsc 2 xdx = cotx CC In cosx tan xdx = =In secx tanx cscx secx tan xdx = secx C csc x cot xdx = cscx 2x 1sin xdx = — 一 sin 2x 2 42」x 1cos xdx = sin2x 2 4n1 n 1 sin xdx = sin xcosx n n | 1 n 1 .cos xdx = cos xsinx n 1 cosx n 1n 1 sin x 1 sin x dx n sin x dxn 丄n 1cos x n 1 cos x m . n | 1cos xsin xdx = m cotx Cn 2sin xdx cos n 2 xdx dx__^"2~sin x dx彳 n 2 n 1 cos x m 1 . n 1 m 1 — cos xsi n x --------------n m n 1 m 1 . n 1 n cos xsi n x m cosm n m n 2. n .xsin xdxm ・ n 2.cos xsin xdx sin ax cosbxdx =cos(a b)xcos(a b)x C2(a b)2(a b)84. 85. 86.87.88.89. 90. 91. 92.93. 94.95. 96. 97.98. 99.100.1sin ax sin bxdx = sin(a b)x2( a b)1cosax cosbxdx = sin(a b)x(a b)dx a bsinx2a2ata n?=arcta n2b2；a2b2dx a bsinx =2l nadx a bcosx1sin (a b)x C2(a b)1sin(a b)x2(a b)b2)atan兰b 4b~a22 ___________atan^ b 4b~a22dxb a bcosxdxa2 cos2 x b2 sin2 x ______ dxa2 cos2 x b2 sin2 x(a2b2)(a2b2)CC1 arctan(— tan x)ab aab丄ln2abxsin axdx = $sin axa2 1 2x sin axdx = x cosaxax cosaxdx = $ cosaxa1x cosaxdx = x sin axa含有反三角函数的积分(其中(abta nx abta nx a1-x cos ax C a2 .2 xs in axa1 . xsi n axa22 xcosaxaa 0)arcsin x dx = xarcsin 仝Ca a23 cosaxa2 .3 sin ax C a101. 102. 103. 104. 105. 106. 107. 10 8. 109.110. 111.112. (十二) 113.X2• X, /X xarcs in dx =(—22X. /X x arccos-dx =(— a 2arctan — dx = x arcta n — a114.115.116.117.118.119. 120.121.(十三)122. 123. 124. 125. 126.127.12 8. 129.x 2arcsin ^dx a 3x . x =—arcs in 3 1 / 29(x2a 2)\ a 2x 2arccos xdx = axx arccos-ax 2arccos —dx a3x =—arc cos —3 a 1(x 2 2a 2). ax 2xarctan x dx = - (a 2 a 2 x 2 )arcta nax C23 x 2 arctan dx = arctan —a 3 x 2)含有指数函数的积分 1 a X dx = ----- a x C In a ax 1 ax e dx = e C a xe dx = 2 (ax a n ax 1 n ax x e dx = x e a ax1)e1 ax .e dx x x xxa dx = aIn a 2a(ln a)n x 1 nx a dx = x In aIn a n 1 x . x a dx e ax s in bxdx =— a e ax(as inbx bcosbx)e ax cosbxdx = b 2 2 -_ e ax (bsinbx acosbx) a b 2a . x)arcsin 4 a x -a x C 42a x)arccos 一 4 a al n(a 2a 2x 2) C144.130. e ax s in nbxdx1ax . n 1ea bnsin bx(a s inbx nbcosbx)131. ax n e cos bxdx =2n(n 1)bb 2n 2ax . n 2 e sin bxdx axbV en cos 1bx(a cosbx nbsin bx)2n(n 1)b ax e cos n 2bxdx 132. In xdx = xlnxx Cdx ,133. -lnln xCxln xn -1n 1八1134. x ln xdx -x (ln x n 1n135. (ln x)n dx - x(ln x)n n (ln x)136. x m(ln x)nd: x -1 mx 1(ln x)r含有对数函数的积分（十四） nm2 .22 1)C1dxmx (lnm 1x)n 1dx （十五）含有双曲函数的积分137.shxdx - chx C13 8. chxdx - shx C139. thxdx - lnchx :C 140. sh 2xdx -- 2 1sh2x C4141.2 x ch xdx -— 2 1 sh2x C4（十六）定积分142.cosnxdx -sin nxdx -0, m cos mxcos nxdx =143 .0 cosmxsin nxdx = 0144.= —C145.sin mxsin nxdx =0, m n146.—,m n20, m nsin mxsin nxdx =cosmxcos nxdx =0 2n = 2nxdx22cos nxdxn 0n 1 In -12nn 1 n 3L 4 2/ 一 亠厶 _4-- ~r\ 4 r^rAr 来、n— n 为大于1旳止可数），I 1 -1 n n 2 5 3 n 1 n 3 3 1（n 为正偶数），I 0 -nL -nn 24 2 22147.2x a b / —、(b a)2x a)(b x)dx =--------------------- 乂x a)(b x) arcsin4 4。