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 C11．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 b - 17．x＝b 18．x＝2a x -+⎰ （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 x x 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．xC34．x＝C +35．2x2ln(2a x C ++36．2x＝ln(x C +++37．1ln aC a x +38．C +39．x2ln(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．d x 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．C +47．x C48．x ＝C +49．2x 2ln 2a x C ++50．2x ＝ln x C +++ 51．1arccos a C a x+52．C +53．x 2ln 2a x C ++54．x ＝2243(25ln 88x x a a x C -+++55．x ⎰C +56．x x ⎰＝422(2ln 88x a x a x C --++57．d x x⎰arccos a a C x +58．2d x x ⎰＝ln x C x -+++(0)a >的积分59．＝arcsin x C a+ 60．C +61．x ＝C62．x C +63．2x ＝2arcsin 2a x C a ++ 64．2x arcsin x C a -+65．1ln a C a x -+66．C +67．x 2arcsin 2a x C a ++68．x ＝2243(52arcsin 88x x a x a C a-+69．x ⎰＝C70．x x ⎰＝422(2arcsin 88x a x x a C a -+71．x a C +72．x ＝arcsin x C a -+(0)a >的积分73．2ax b C +++74．x75．x 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 a C - （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 +⎰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 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--+ 131．e cos d ax n bx x ⎰＝12221e cos (cos sin )ax n bx a bx nb bx 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。