高等数学常用积分公式总结大全目录高等数学常用积分公式总结大全 (1)（一）含有ax b +的积分(0a ≠) .............................................................................................. 2 （二）含有ax b +的积分 . (2)（三）含有22x a ±的积分 (3)（四）含有2(0)ax b a +>的积分 (4)（五）含有2ax bx c ++(0)a >的积分 ................................................................................... 4 （六）含有22x a +(0)a >的积分 ........................................................................................ 5 （七）含有22x a −(0)a >的积分 ........................................................................................ 6 （八）含有22a x −(0)a >的积分 ........................................................................................ 7 （九）含有2ax bx c ±++(0)a >的积分 ............................................................................. 8 （十）含有x a x b−±−或()()x a b x −−的积分 .................................................................... 9 （十一）含有三角函数的积分 (9)（十二）含有反三角函数的积分（其中0a >） (11)（十三）含有指数函数的积分 (12)（十四）含有对数函数的积分 (13)（十五）含有双曲函数的积分 (13)（十六）定积分 (14)（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+−++ （二）含有ax b +的积分 10．d ax b x +⎰＝32()3ax b C a++ 11．d x ax b x +⎰＝322(32)()15ax b ax b C a −++12．2d xax b x +⎰＝222332(15128)()105a x abx b ax b C a −+++ 13．d x x ax b+⎰＝22(2)3ax b ax b C a −++ 14．2d x x ax b+⎰＝22232(348)15a x abx b ax b C a −+++ 15．d x x ax b +⎰＝1ln (0)2arctan (0)ax b b C b b ax b b ax b C b b b ⎧+−+>⎪++⎪⎨⎪++<⎪−−⎩16．2d x xax b +⎰＝d 2ax b a x bx b x ax b +−−+⎰ 17．d ax b x x +⎰＝d 2x ax b b x ax b +++⎰ 18．2d ax b x x +⎰＝d 2ax b a x x x ax b +−++⎰ （三）含有22xa ±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 +⎰＝1arctan (0)1ln(0)2ax C b b ab ax b C b ab ax b ⎧+>⎪⎪⎨−−⎪+<⎪−+−⎩23．2d x x ax b +⎰＝21ln 2ax b C a ++24．22d x x ax b +⎰＝2dx b xa a axb −+⎰25．2d ()x x ax b +⎰＝221ln 2x C b ax b ++26．22d ()xx ax b +⎰＝21d a xbx b ax b −−+⎰27．32d ()x x ax b +⎰＝22221ln 22ax b aC b x bx +−+28．22d ()xax 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 ++⎰＝222222222arctan (4)44124ln (4)424ax b Cb ac ac b ac bax b b acCb ac b ac ax b b ac +⎧+<⎪−−⎪⎨+−−⎪+>⎪−++−⎩ 30．2d xx ax bx c ++⎰＝221d ln 22b xax bx c a a ax bx c++−++⎰（六）含有22x a +(0)a >的积分 31．22d xx a +⎰＝1arsh x C a+＝22ln()x x a C +++ 32．223d ()x x a +⎰＝222x C a x a ++ 33．22d xx x a +⎰＝22x a C ++34．223d ()x x x a +⎰＝221C x a −++35．222d x x x a +⎰＝22222ln()22x a x a x x a C +−+++ 36．2223d ()x x x a +⎰＝2222ln()x x x a C x a −+++++ 37．22d x x x a+⎰＝221ln x a a C a x +−+ 38．222d xx x a +⎰＝222x a C a x +−+ 39．22d x a x +⎰＝22222ln()22x a x a x x a C +++++ 40．223()d x a x +⎰＝22224223(25)ln()88x x a x a a x x a C ++++++ 41．22d x x a x +⎰＝2231()3x a C ++ 42．222d x x a x +⎰＝4222222(2)ln()88x a x a x a x x a C ++−+++43．22d x a x x +⎰＝2222ln x a a x a a C x +−+++ 44．222d x a x x +⎰＝2222ln()x a x x a C x+−++++ （七）含有22x a −(0)a >的积分 45．22d x x a −⎰＝1arch x x C x a +=22ln x x a C +−+ 46．223d ()x x a −⎰＝222x C a x a −+− 47．22d xx x a −⎰＝22x a C −+48．223d ()x x x a −⎰＝221C x a −+−49．222d x x x a −⎰＝22222ln 22x a x a x x a C −++−+ 50．2223d ()x x x a −⎰＝2222ln x x x a C x a −++−+− 51．22d x x x a −⎰＝1arccos a C a x+ 52．222d x xx a −⎰＝222x a C a x −+ 53．22d x a x −⎰＝22222ln 22x a x a x x a C −−+−+ 54．223()d x a x −⎰＝22224223(25)ln 88x x a x a a x x a C −−++−+55．22d x x a x −⎰＝2231()3x a C −+ 56．222d x x a x −⎰＝4222222(2)ln 88x a x a x a x x a C −−−+−+ 57．22d x a x x −⎰＝22arccos a x a a C x −−+ 58．222d x a x x −⎰＝2222ln x a x x a C x −−++−+ （八）含有22a x −(0)a >的积分 59．22d xa x −⎰＝arcsin x C a+ 60．223d ()x a x −⎰＝222x C a a x +− 61．22d xx a x −⎰＝22a x C −−+62．223d ()x x a x −⎰＝221C a x +−63．222d x x a x −⎰＝222arcsin 22x a x a x C a −−++ 64．2223d ()x x a x −⎰＝22arcsin x x C a a x −+− 65．22d x x a x−⎰＝221ln a a x C a x −−+ 66．222d x x a x −⎰＝222a x C a x −−+67．22d a x x −⎰＝222arcsin 22x a x a x C a −++ 68．223()d a x x −⎰＝222243(52)arcsin 88x x a x a x a C a−−++ 69．22d x a x x −⎰＝2231()3a x C −−+ 70．222d x a x x −⎰＝42222(2)arcsin 88x a x x a a x C a −−++ 71．22d a x x x −⎰＝2222ln a a x a x a C x −−−++ 72．222d a x x x −⎰＝22arcsin a x x C x a−−−+ （九）含有2ax bx c ±++(0)a >的积分 73．2d xax bx c ++⎰＝21ln 22ax b a ax bx c C a+++++ 74．2d ax bx c x ++⎰＝224ax b ax bx c a+++2234ln 228ac b ax b a ax bx c C a −++++++ 75．2d xx ax bx c ++⎰＝21ax bx c a++ 23ln 222b ax b a ax bx c C a −+++++ 76．2d xc bx ax +−⎰＝212arcsin 4ax b C a b ac−−++77．2d c bx ax x +−⎰＝2232242arcsin 484ax b b ac ax b c bx ax C a a b ac −+−+−+++ 78．2d xx c bx ax +−⎰＝23212arcsin 24b 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 −−⎰＝22()()()arcsin 44x 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 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 tan 2x 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 n x 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 n x 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 +⎰＝2222tan 22arctan x a b C a b a b ++−−22()a b >104．d sin x a b x +⎰＝222222tan12ln tan 2x a b b a C x b a a b b a +−−+−++−22()a b < 105．d cos x a b x +⎰＝2arctan(tan )2a b a b x C a b a b a b +−++−+22()a b > 106．d cos x a b x +⎰＝tan 12ln tan 2x a ba b b a C a b b a x a bb a +++−++−+−−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 xx a ⎰＝22arcsin x x a x C a+−+ 114．arcsin d x x x a⎰＝2222()arcsin 244x a x x a x C a −+−+115．2arcsin d x x x a ⎰＝322221arcsin (2)39x x x a a x C a ++−+ 116．arccos d xx a ⎰＝22arccos x x a x C a−−+ 117．arccos d x x x a⎰＝2222()arccos 244x a x x a x C a −−−+ 118．2arccos d x x x a ⎰＝322221arccos (2)39x x x a a x C a −+−+ 119．arctand 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 xa x ⎰＝1ln x a C a+ 123．e d ax x ⎰＝1e ax C a+ 124．e d ax x x ⎰＝21(1)e ax ax C a−+ 125．e d n ax x x ⎰＝11e e d n ax n ax n x x x a a −−⎰ 126．d x xa 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 ax bx x ⎰＝221e (sin cos )ax a bx b bx C a b−++ 129．e cos d ax bx x ⎰＝221e (sin cos )ax b bx a bx C a b +++130．e sin d ax n bx 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 n x x x ⎰＝111(ln )11n x x C n n +−+++ 135．(ln )d n x x ⎰＝1(ln )(ln )d n n x x n x x −−⎰ 136．(ln )d m n x 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⎰＝0 143．cos sin d mx nx x π−π⎰＝0 144．cos cos d mx nx x π−π⎰＝0,,m n m n≠⎧⎨π=⎩ 145．sin sin d mx nx x π−π⎰＝0,,m n m 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 n x x π⎰＝20cos d n x x π⎰ n I ＝21n n I n−− 1342253n n n I n n −−=⋅⋅⋅⋅− （n 为大于1的正奇数），1I ＝1 13312422n n n I n n −−π=⋅⋅⋅⋅⋅−（n 为正偶数），0I ＝2π。