创作编号：GB8878185555334563BT9125XW创作者： 凤呜大王*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 ()xx ax b +⎰＝1ln ax b C b x +-+6．2d ()xx 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 ()xx 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 ⎰＝b ⎰18．2d x x ⎰＝2a + （三）含有22x a ±的积分 19．22d x x a +⎰=1arctan x C a a+20．22d ()nx x a +⎰=2221222123d 2(1)()2(1)()n n x n xn a x a n a x a ---+-+-+⎰21．22d xx 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 xbx b ax b --+⎰27．32d ()x x ax b +⎰＝22221ln 22ax b a C b x bx+-+ 28．22d ()x ax b +⎰＝221d 2()2x xb 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)C b acC b ac+<+>30．2dxxax bx c++⎰＝221dln22b xax bx ca a ax bx c++-++⎰(0)a>的积分31．⎰＝1arshxCa+＝ln(x C++32．C+33．x⎰C34．x＝C+35．2x2ln(2ax C++36．2x⎰＝ln(x C++37．⎰1lnaCa x-+38．⎰C+39．x⎰2ln(2ax C++40．x⎰＝2243(25ln(88x x a a x C +++41．x ⎰C42．xx⎰＝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 xC x a+=ln x C ++ 46．⎰C +47．x ⎰C +48．x ＝C +创作编号：GB8878185555334563BT9125XW创作者： 凤呜大王*49．2x 2ln 2a x C ++50．2x ⎰＝ln x C +++51．⎰＝1arccos aC a x+52．⎰C +53．x ⎰2ln 2a x C ++54．x⎰＝2243(25ln 88x x a a x C -++55．x ⎰C56．xx⎰＝422(2ln 88x a x a x C -++57．x ⎰arccos a a C x +58．x ⎰＝ln x C +++(0)a >的积分59．⎰＝arcsinxC a + 60．⎰C +61．x ⎰＝C62．x C +63．2x ＝2arcsin 2a x C a + 64．2x ⎰arcsinxC 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 ⎰＝C +70．xx ⎰＝422(2arcsin 88x a x x a C a-+71．d x x ⎰a C ++72．x ⎰＝arcsin xC a-+(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()arcsin 4b a C -+()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()42xC π++＝ln sec tan x x C ++ 88．csc d x x ⎰＝ln tan2xC +＝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 xn x n x----⋅+--⎰ 98．d cos n x x ⎰＝121sin 2d 1cos 1cos n n x n xn x n x---⋅+--⎰ 99．cos sin d m n x x x ⎰＝11211cos sin cos sin d m n m nm 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 xa b x +⎰tanxa b C ++22()a b >104．d sin x a b x+⎰＝C+22()a b <105．d cos x a b x+⎰＝)2xC+22()a b >106．d cos xa b x+⎰＝C +22()a b <创作编号：GB8878185555334563BT9125XW创作者： 凤呜大王*107．2222d cos sin x a x b x +⎰＝1arctan(tan )bx C ab a+ 108．2222d cos sin xa xb 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 xx 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 xx a ⎰＝arccosxx C a-117．arccos d xx x a ⎰＝22()arccos 24x a x C a -118．2arccos d x x x a⎰＝3221arccos (239x x x a 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 xx 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 xa C a + 123．e d axx ⎰＝1e ax C a +124．e d ax x x ⎰＝21(1)e axax 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 nxx a x ⎰＝11d ln ln n x n xn 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 ax bx x ⎰＝221e (sin cos )axb 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 nbx 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 xx x ⎰＝ln ln x C +134．ln d nx x x ⎰＝111(ln )11n x x C n n +-+++ 135．(ln )d n x x ⎰＝1(ln )(ln )d n nx 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．sin 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 --=⋅⋅⋅⋅- （n 为大于1的正奇数），1I ＝113312422n n n I n n --π=⋅⋅⋅⋅⋅-（n 为正偶数），0I ＝2π创作编号：GB8878185555334563BT9125XW创作者：凤呜大王*。