常 用 积 分 公 式（一）含有ax b +的积分(0a ≠)1．d x ax b +⎰＝1ln ax b C a ++2．()d ax b x μ+⎰＝11()(1)ax b C a μμ++++（1μ≠-） 3．d x x ax b +⎰＝21(ln )ax b b ax b C a +-++4．2d x x ax b +⎰＝22311()2()ln 2ax b b ax b b ax b C a ⎡⎤+-++++⎢⎥⎣⎦ 5．d ()x x ax b +⎰＝1ln ax b C b x +-+6．2d ()x x ax b +⎰＝21ln a ax b C bx b x +-++7．2d ()x x ax b +⎰＝21(ln )b ax b C a ax b ++++8．22d ()x x ax b +⎰＝231(2ln )b ax b b ax b C a ax b +-+-++9．2d ()x x ax b +⎰＝211ln ()ax b C b ax b b x +-++的积分10．x⎰＝C 11．x ⎰＝22(3215ax b C a -+12．x x ⎰＝22232(15128105a x abx b C a -+ 13．x ⎰＝22(23ax b C a - 14．2x ⎰＝22232(34815a x abx b C a -+ 15．⎰(0)(0)C b C b ⎧+>+<16．⎰＝2a bx b -- 17．x ⎰＝b18．x ⎰＝2a +（三）含有22x a ±的积分19．22d x x a +⎰=1arctan x C a a +20．22d ()n x x a +⎰=2221222123d 2(1)()2(1)()n n x n x n a x a n a x a ---+-+-+⎰21．22d x x a -⎰=1ln 2x a C a x a -++（四）含有2(0)ax b a +>的积分 22．2d x ax b +⎰＝(0)(0)C b C b ⎧+>+<23．2d x x ax b +⎰＝21ln 2ax b C a ++24．22d x x ax b +⎰＝2d x b x a a ax b -+⎰25．2d ()x x ax b +⎰＝221ln 2x C b ax b ++26．22d ()x x ax b +⎰＝21d a x bx b ax b --+⎰27．32d ()x x ax b +⎰＝22221ln 22ax b a C b x bx +-+28．22d ()x ax b +⎰＝221d 2()2xxb ax b b ax b +++⎰（五）含有2ax bx c ++(0)a >的积分29．2d x ax bx c ++⎰＝22(4)(4)Cb ac Cb ac +<+> 30．2d xx ax bx c ++⎰＝221d ln 22b xax bx c a a ax bx c++-++⎰(0)a >的积分31．⎰＝1arsh xC a +＝ln(x C ++32．⎰C +33．x ⎰C34．x ⎰＝C +35．2x ⎰2ln(2a x C ++36．＝ 37．⎰1ln a C a x -+38．⎰C +39．x ⎰2ln(2a x C ++ 40．x ⎰＝2243(25ln(88x x a a x C ++++ 41．x ⎰C + 42．x x ⎰＝422(2ln(88x a x a x C +++43．x ⎰ln a a C x + 44．2d x x ⎰＝ln(x C x -+++(0)a >的积分45．＝1arch x x C x a +=ln x C ++46．47．x⎰C48．x⎰＝C+49．2x⎰2ln2ax C+++50．2x⎰＝ln x C+++51．⎰1arccos a Ca x+52．⎰C+53．x2ln2ax C++54．x⎰＝2243(25ln88xx a a x C-+++55．x⎰C56．x x⎰＝422(2ln88x ax a x C-++57．x⎰arccosaa Cx+58．x⎰＝ln x C+++(0)a>的积分59．⎰＝arcsinxCa+60．⎰C+61．x⎰＝C62．x⎰C+63．2x⎰＝2arcsin2a xCa+64．2x⎰arcsinxCa-+65．⎰1lnaCa x-+66．⎰2Ca x-+67．x2arcsin2a xCa+68．x⎰＝2243(52arcsin88x xa x a Ca-+69．x⎰＝C70．x x⎰＝422(2arcsin88x a xx a Ca-++71．d xx⎰a C+72．x⎰＝arcsinxCa-+(0)a>的积分73．⎰2ax b C+++74．x⎰22ax b C++++75．x ⎰＝2ax b C-+++76．⎰＝C+77．x⎰2C+ 78．x⎰＝C++79．x⎰＝((x b b a C--+80．x⎰＝((x b b a C-+-+81．⎰C()a b<82．x⎰2()4b aC-()a b<（十一）含有三角函数的积分83．sin d x x⎰＝cos x C-+84．cos d x x ⎰＝sin x C +85．tan d x x ⎰＝ln cos x C -+86．cot d x x ⎰＝ln sin x C +87．sec d x x ⎰＝ln tan()42x C π++＝ln sec tan x x C ++88．csc d x x ⎰＝ln tan2x C +＝ln csc cot x x C -+ 89．2sec d x x ⎰＝tan x C + 90．2csc d x x ⎰＝cot x C -+91．sec tan d x x x ⎰＝sec x C +92．csc cot d x x x⎰＝csc x C -+ 93．2sin d x x ⎰＝1sin 224x x C -+ 94．2cos d x x ⎰＝1sin 224x x C ++ 95．sin d n x x ⎰＝1211sin cos sin d n n n x x x x n n ----+⎰96．cos d nx x ⎰＝1211cos sin cos d n n n x x x x n n ---+⎰97．d sin n x x ⎰＝121cos 2d 1sin 1sin n n x n x n x n x ----⋅+--⎰98．d cos n x x ⎰＝121sin 2d 1cos 1cos n n x n x n x n x ---⋅+--⎰99．cos sin d m nx x x ⎰＝11211cos sin cos sin d m n m n m x x x x x m n m n -+--+++⎰ ＝11211cos 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 ⎰＝11cos()cos()2()2()a b x a b x C a b a b -+--++- 101．sin sin d ax bx x ⎰＝11sin()sin()2()2()a b x a b x C a b a b -++-++- 102．cos cos d ax bx x ⎰＝11sin()sin()2()2()a b x a b x C a b a b ++-++-103．d sin x a b x +⎰tan x a b C ++22()a b >104．d sin x a b x +⎰C +22()a b < 105．d cos x a b x +⎰＝)2x C +22()a b >106．d cos x a b x +⎰＝C +22()a b <107．2222d cos sin x a x b x +⎰＝1arctan(tan )b x C ab a +108．2222d cos sin x a x b x -⎰＝1tan ln 2tan b x a C ab b x a ++-109．sin d x ax x ⎰＝211sin cos ax x ax C a a -+110．2sin d x ax x ⎰＝223122cos sin cos x ax x ax ax C a a a -+++ 111．cos d x ax x ⎰＝211cos sin ax x ax C a a ++112．2cos d x ax x ⎰＝223122sin cos sin x ax x ax ax C a a a +-+ （十二）含有反三角函数的积分（其中0a >）113．arcsin d x x a ⎰＝arcsin x x C a +114．arcsin d x x x a ⎰＝22()arcsin 24x a x C a -+115．2arcsin d x x x a ⎰＝3221arcsin (239x x x a C a ++116．arccos d x x a ⎰＝arccos x x C a -117．arccos d x x x a ⎰＝22()arccos 24x a x C a -118．2arccos d x x x a ⎰＝3221arccos (239x x x a C a -++ 119．arctan d x x a ⎰＝22arctan ln()2x a x a x C a -++120．arctan d x x x a ⎰＝221()arctan 22x a a x x C a +-+121．2arctan d x x x a ⎰＝33222arctan ln()366x x a a x a x C a -+++ （十三）含有指数函数的积分122．d x a x⎰＝1ln x a C a + 123．e d axx ⎰＝1e ax C a + 124．e d axx x ⎰＝21(1)e ax ax C a -+ 125．e d n axx x ⎰＝11e e d n ax n ax n x x x a a --⎰126．d xxa x ⎰＝21ln (ln )x x x a a C a a -+ 127．d n x x a x ⎰＝11d ln ln n x n x n x a x a x a a --⎰128．e sin d axbx x ⎰＝221e (sin cos )ax a bx b bx C a b -++ 129．e cos d axbx x ⎰＝221e (sin cos )ax b bx a bx C a b +++ 130．e sin d ax nbx x ⎰＝12221e sin (sin cos )ax n bx a bx nb bx a b n --+ 22222(1)e sin d ax n n n b bx x a b n --++⎰131．e cos d ax n bx x ⎰＝12221e cos (cos sin )ax n bx a bx nb bx a b n -++ 22222(1)e cos d ax n n n b bx x a b n --++⎰（十四）含有对数函数的积分132．ln d x x ⎰＝ln x x x C -+133．d ln x x x ⎰＝ln ln x C +134．ln d nx x x ⎰＝111(ln )11n x x C n n +-+++ 135．(ln )d nx x ⎰＝1(ln )(ln )d n n x x n x x --⎰136．(ln )d m nx x x ⎰＝111(ln )(ln )d 11m n m n n x x x x x m m +--++⎰ （十五）含有双曲函数的积分137．sh d x x ⎰＝ch x C + 138．ch d x x ⎰＝sh x C + 139．th d x x ⎰＝lnch x C +140．2sh d x x ⎰＝1sh224x x C -++ 141．2ch d x x ⎰＝1sh224x x C ++ （十六）定积分142．cos d nx x π-π⎰＝sin d nx x π-π⎰＝0143．cos sin d mx nx x π-π⎰＝0144．cos cos d mx nx x π-π⎰＝0,,m nm n ≠⎧⎨π=⎩ 145．sin sin d mx nx x π-π⎰＝0,,m nm n ≠⎧⎨π=⎩ 146．0sin sin d mx nx x π⎰＝0cos cos d mx nx x π⎰＝0,,2m n m n ≠⎧⎪⎨π=⎪⎩147． n I ＝20sin d nx x π⎰＝20cos d n x x π⎰n I ＝21n n I n --1342253n n n I n n --=⋅⋅⋅⋅-L （n 为大于1的正奇数），1I ＝113312422n n n I n n --π=⋅⋅⋅⋅⋅-L （n 为正偶数），0I ＝2π。