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高等数学积分公式大全

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 +⎰=1lnax 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 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 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 bxc ++(0)a >的积分29.2d x ax bx c ++⎰=22(4)(4)C b ac Cb ac +<+>30.2d x x ax bx c ++⎰=221d ln 22b x ax 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 ⎰C42.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 xC x a+=ln x C ++ 46.C +47.x C +48.x =C +49.2x 2ln 2a x C ++50.2x =ln x C +++51.1arccosaC ax+52.C +53.x 2ln 2a x C ++54.x =2243(25ln 88x x a a x C -+++55.x ⎰C56.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 xC 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.x x ⎰=422(2arcsin 88x a x x a C a-+71.x a C ++72.x =arcsin xC 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 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 xa 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 )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 xx a ⎰=arcsin x x C a+114.arcsin d xx x a ⎰=22()arcsin 24x a x C a -+115.2arcsin d xx x a⎰=3221arcsin (239x x x a C a +++116.arccos d x x a ⎰=arccos x x C a117.arccos d xx x a ⎰=22()arccos 24x a x C a --+118.2arccos d xx x a⎰=3221arccos (239x x x a C a -++119.arctan d xx 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 x a x ⎰=1ln xa 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 nx x x a a--⎰126.d x xa x ⎰=21ln (ln )x xx a a C a a -+ 127.d n x x a x ⎰=11d ln ln n x n xn 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 xx 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 nx 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 ⎰=1sh224xx 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⎰=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 =1 13312422n n n I n n --π=⋅⋅⋅⋅⋅-L(n 为正偶数),0I =2π。

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