第一部分:常用积分公式基本积分公式: 1kdx kx c =+⎰2 11x x dx c μμμ+=++⎰ 3ln dxx c x =+⎰4 ln xxa a dx c a=+⎰ 5 x xe dx ec =+⎰6 cos sin xdx x c =+⎰7 sin cos xdx x c =-+⎰8221sec tan cos dx xdx x c x ==+⎰⎰9 221csc cot sin xdx x c x ==-+⎰⎰ 10 21arctan 1dx x c x=++⎰ 11arcsin x c =+12 tan ln cos xdx x c =-+⎰ 13 cot ln sin xdx x c =+⎰14 sec ln sec tan xdx x x c =++⎰ 15 csc ln csc cot xdx x x c =-+⎰162211arctan xdx c a x a a=++⎰ 172211ln 2x a dx c x a a x a-=+-+⎰ 18arcsinxc a=+19ln x c =+分部积分法公式1 形如n ax x e dx ⎰,令n u x =,axdv e dx =2 形如sin n x xdx ⎰令nu x =,sin dv xdx =3 形如cos n x xdx ⎰令nu x =,cos dv xdx = 4 形如arctan n x xdx ⎰,令arctan u x =,ndv x dx =5 形如ln n x xdx ⎰,令ln u x =,ndv x dx =6 形如sin ax e xdx ⎰,cos ax e xdx ⎰令,sin ,cos axu e x x =均可。
常用凑微分公式 1. ()()()1f ax b dx f ax b d ax b a+=++⎰⎰ 2.()()()11f x x dx f x d x μμμμμ-=⎰⎰3. ()()()1ln ln ln f x dx f x d x x⋅=⎰⎰4. ()()()x x x x f e e dx f e d e ⋅=⎰⎰5. ()()()1ln x x x x f a a dx f a d a a⋅=⎰⎰ 6. ()()()sin cos sin sin f x xdx f x d x ⋅=⎰⎰ 7. ()()()cos sin cos cos f x xdx f x d x ⋅=-⎰⎰ 8.()()()2tan sectan tan f x xdx f x d x ⋅=⎰⎰9.2dx f d=⎰ 10.21111()()()f dx f d x x x x =-⎰⎰11.()()()2cot csccot cot f x xdx f x d x ⋅=⎰⎰第二部分:常用微分、导数公式(c=常数)1、极限(1)0sin lim 1x xx→= (2)()10lim 1x x x e →+= (3))1n a o >=(4)1n = (5)limarctan 2x x π→∞=(6)lim tan 2x arc x π→-∞=-(7)limarccot 0x x →∞= (8)lim arccot x x π→-∞= (9)lim 0x x e →-∞=(10)lim x x e →+∞=∞ (11)0lim 1xx x +→= (12)0101101lim 0n n n m m x m a n mb a x a x a n m b x b x b n m--→∞⎧=⎪⎪+++⎪=<⎨+++⎪∞>⎪⎪⎩L L (系数不为0的情况) (13)000()()limx x x xf x f x y x →+∆-∆=∆∆2、常用等价无穷小关系(0x →)sin ~x x tan ~x x arcsin ~x x arctan ~x x 211cos ~2x x - ()ln 1~x x + 1~x e x - 1~ln x a x a -()11~x x ∂+-∂ 21sec 1~2x x -211~2x2~x33sin ~()x x3、导数的四则运算法则()u v u v '''±=± ()uv u v uv '''=+ 2u u v uv v v '''-⎛⎫= ⎪⎝⎭4、基本导数公式⑴()0c '= ⑵1x x μμμ-= ⑶()sin cos x x '= ⑷()cos sin x x '=- ⑸()2tan sec x x '= ⑹()2cot csc x x '=- ⑺()sec sec tan x x x '=⋅ ⑻()csc csc cot x x x '=-⋅⑼()x x e e '= ⑽()ln x x a a a '= ⑾()1ln x x'=⑿()1log ln x a x a '=⒀()arcsin x '= ⒁()arccos x '=⒂()21arctan 1x x '=+ ⒃()21arccot 1x x '=-+⒄()1x '=⒅'=5、高阶导数的运算法则 (1)()()()()()()()n n n u x v x u x v x ±=±⎡⎤⎣⎦ (2)()()()()n n cu x cu x =⎡⎤⎣⎦(3)()()()()n n nu ax b a uax b +=+⎡⎤⎣⎦(4)()()()()()()()0nn n k k k n k u x v x c u x v x -=⋅=⎡⎤⎣⎦∑ 6、基本初等函数的n 阶导数公式 (1)()()!n n x n = (2)()()n ax b n ax b e a e ++=⋅ (3)()()ln n x x n a a a =(4)()()sin sin 2n n ax b a ax b n π⎛⎫+=++⋅⎡⎤ ⎪⎣⎦⎝⎭(5) ()()cos cos 2n n ax b a ax b n π⎛⎫+=++⋅⎡⎤ ⎪⎣⎦⎝⎭(6)()()()11!1n n nn a n ax b ax b +⋅⎛⎫=- ⎪+⎝⎭+ (7)()()()()()11!ln 1n n n na n axb ax b -⋅-+=-⎡⎤⎣⎦+7、微分公式与微分运算法则⑴()0d c = ⑵()1d x x dx μμμ-= ⑶()sin cos d x xdx = ⑷()cos sin d x xdx =- ⑸()2tan sec d x xdx = ⑹()2cot csc d x xdx =- ⑺()sec sec tan d x x xdx =⋅ ⑻()csc csc cot d x x xdx =-⋅ ⑼()x x d e e dx = ⑽()ln x x d a a adx = ⑾()1ln d x dx x= ⑿()1log ln x a d dx x a =⒀()arcsin d x =⒁()arccos d x =⒂()21arctan 1d x dx x =+ ⒃()21arccot 1d x dx x=-+8、微分运算法则⑴()d u v du dv ±=± ⑵()d cu cdu =⑶()d uv vdu udv =+ ⑷2u vdu udvd v v -⎛⎫= ⎪⎝⎭第三部分:常用三角函数公式1.和差公式sin()sin cos cos sin A B A B A B +=+ sin()sin cos cos sin A B A B A B -=- cos()cos cos sin sin A B A B A B +=- cos()cos cos sin sin A B A B A B -=+tan tan tan()1tan tan A B A B A B ++=- tan tan tan()1tan tan A BA B A B --=+cot cot 1cot()cot cot A B A B B A ⋅-+=+ cot cot 1cot()cot cot A B A B B A ⋅+-=- 2.倍角公式sin 22sin cos A A A = 2222cos 2cos sin 12sin 2cos 1A A A A A =-=-=- 22tan tan 21tan AA A=- 3.半角公式sin2A = cos 2A =sin tan21cos A A A ==+ sin cot 21cos A A A==- 4.和差化积公式sin sin 2sincos 22a b a b a b +-+=⋅ sin sin 2cos sin22a b a ba b +--=⋅ cos cos 2cos cos 22a b a b a b +-+=⋅ cos cos 2sin sin 22a b a ba b +--=-⋅()sin tan tan cos cos a b a b a b++=⋅5.积化和差公式()()1sin sin cos cos 2a b a b a b =-+--⎡⎤⎣⎦()()1cos cos cos cos 2a b a b a b =++-⎡⎤⎣⎦()()1sin cos sin sin 2a b a b a b =++-⎡⎤⎣⎦ ()()1cos sin sin sin 2a b a b a b =+--⎡⎤⎣⎦6.万能公式22tan2sin 1tan 2aa a=+ 221tan 2cos 1tan 2a a a -=+ 22tan2tan 1tan 2aa a=- 7.平方关系22sin cos 1x x += 22sec n 1x ta x -= 22csc cot 1x x -=8.倒数关系tan cot 1x x ⋅= sec cos 1x x ⋅= c sin 1cs x x ⋅=9.商数关系sin tan cos x x x =cos cot sin xx x= 10.正弦定理:R C cB b A a 2sin sin sin ===11.余弦定理:C ab b a c cos 2222-+= 12.反三角函数性质:arcctgx arctgx x x -=-=2arccos 2arcsin ππ。