大學(xué)大全整理0001

1、+b°0（系數(shù)不為0的情況）二、重要公式（1）lim= 1XTO(3) lim亦(a >o) = l(4) lim亦=1(5)T8liinarctaiix = 2A->007t lmi arc tan x =2(7) limarccotx = 0(8) lun aiccotx=(9).tTX(10) liin ex = oo(11) lim xr = 1x->0*三、下列常用等價無窮小關(guān)系sinx xtanx xaicsinx xarctanx 兀l-COSX- ix22ln(l + x)xer -1 x四、導(dǎo)數(shù)的四則運算法則五、基本導(dǎo)數(shù)公式(cj = O(2) #

2、=“嚴(yán)(3)(sinx) =cosx(4)(cosxj = - sin x(5)(tanx) = sec2 x(6)(cotx) =-csc2 X(7)(secx) =secxtanA(8)(cscx) =-cscxcotx("j “f(10)(67 J =axna(ll)(lnx)'=丄f1(13)(arcsinx) = _2yjl-x2:G4)(arccosx)=(15)(arctanx)=】,G6)(arccotx) =_】【、(17)(兀)'=1 (18)(妖)(HV)= U'V+UV(M± V)=ll' ±V'六、

3、高階導(dǎo)數(shù)的運算法則VI-x212yfx1) M(x)±v(x)(n) =M(x)(n)±v(x)(n)(2) cw(x)(w)=cw(n)(x)(3) “(ox+b)何=q”汕)(ox+b)(4) «(%) v(x)(n) =k=0七、基本初等函數(shù)的n階導(dǎo)數(shù)公式（*）"" =川（2）（嚴(yán)）（"）=* 嚴(yán)冗、2丿(4) sin(ox + b)"" = a" sin| ax + b + n- cos(ox+b)"" = a" cos(ax+b + n 彳 禽）=（-1）n（

4、77;八、微分公式與微分運算法則 d（c） = 0 J（xA） = /ixdx班 CQSx) = -snxdx(5) J(tanx) = sec2 xdx(3) d (sin x) = cos xdx (cot x) = - esc2 xdx d(ex>j = exdx(10) d (&') = axnadx(11) d (In x)=丄(7) d (sec x)二 sec x tan xdx(8) d (esc x) = - esc x cot xdx(12)d(log.;') = J dx(13)d (aicsiiix)=】 dx (14)d (arccos

5、x)=.dx、7 xnay/i-x2yji_x2(16) d (arc cot x) = -】,dx(15) d (aictaii x)=】,dx九、微分運算法則(1) d (u 土 v) = du ± dv(3) d (mv) = vdu + udv十、基本積分公式<l)kdx = kx+c dcu) = cdu伯=牛空a" f axdx =+ c(5) f exdx = eInaJx+c(6) JcosAd = sinx+c(7) J sin xdx = - cos x+c(8)f dx= f sec2 xdx = tail x+c 丿 COS" xJ(

6、9) f= fcsc2 Az/x = -cotx+c(10)f dx = aictaii x+c J l + x(11) f idx = arcsinx+c十二、補充下面幾個積分公式 j tail xdx = -lii |cosJsecAzZx = ln|secx4-taiiAj + cJcotAz£v = lii|sinx| j esc xdx = In |csc x - cot x + cI1-X2十一、下列常用湊微分公式積分型換元公式f (ax+bbc =丄 J f (ax+bd (ax+b)u = ax+b“a”艸u = xj/(lnx)= J/(lnx)rf(lnx)Xu

7、= nxu = ex”的川毎需”仗陽')u = axj/(sinx)-cos xdx = j/(siii x)d (sin x)u = smxj f (cos x) sin xdx = -j/ (cos x)d (cos x)ll = cos Xj/(tanx)sec2= j/(tanx)t/(tanx)u = tanxj f (cot x) esc2 xdx = j/(cot x)d (cot x)u = cot Xj/(arctailx)】,dx = f (arctanx)d(arctanx)M = aictanx|/(aicsinx)】= jf (arcsinx)d(arcsii

8、ix)y/l X2u = arcsin 牙f_7 dx = L arctail + cf , 1 . Jx = -lnx-aJ a + xaaJ x -a2ax+a+ c1,. xax = aicsin + c 2-x2a1±a2dx = lii x+>Jx2 ±a2 + c十三、分部積分法公式形如 J x'edx,令 u = xn,dv = edx形如 sinxdx 令“ =/dv = snxdx形如 J xn cos xdx 令“ =xn,dv = QQsxdx形如 ja/' aictaiixdx,令h = aictanx, dv = xndx形如

9、 jxn nxdx,令w = liix, dv = x"dx 形如 J 嚴(yán) sin xdx, j eax cos xdx 令“ =eax, sin x, cos x 均可。十四、第二換元積分法中的三角換元公式(1) yja2 -x2x = asint (2) y/a2 +x2x = aXant(3) yx2 -a2 x=asect【特殊角的三角函數(shù)值】(1)sinO = O(2).7t sin =612.7t(3) sm =32(4) sin2=1)(5) sin = 0(1)cosO = 1(2)n cos =更,、71(3) cos _ 1(4) cos=0 )(5) cos”

10、= -l623_ 22(1)tan0 = 0(2)tan=羽7t(3) tail = >/371(4) tan 不存在(5) tan = 06332(1)cot 0不存在(2)7C cot =5/3(3) cot = (4) cot(5) cot龍不存6332十五、三角函數(shù)公式1. 兩角和公式sin(A -B) = sin A cos B - cos A sill Bsiii(A + B) = sill A cos B + cos A sill Bcos(A + B) = cos A cos B-sinA sin Bcos(A-B) = cos A cos B + sin A sin B

11、tan(A + B) =tan A + tail B1-tail A tail Btan(A -B) =tail A - tail B1 + tan A tail Bcot(A + B) =cot A cot B -1cot B + cot Acot(A 一 B)=cot A cot B +1cot B - cot A2. 二倍角公式sin 2 A = 2 sin A cos Acos2A = cos2 A-sin2 A = l-2siir A = 2cos2 A-ltaii2A =2 tan Al-tan2AA /1-cos Asin Atan = J=2 v 1 + cos A 1+cos

12、 AA /1 + cos Asin Acot = J=2 v 1-cos A 1-cos A4. 和差化積公式a-bI ca + bsina + sinb = 2sin22. - a+ba-bcosa + cos彷=2cos2coscos., -a + b . a-bsin a sin Q = 2 cossm2 2.-.a + b . a-bcos a 一 cos b = -2 smsin2, sin(a + b)tan a + tan b =-cos a cos b5. 積化和差公式sinasinb = 一扌cos(a + b)-cos(a-b) sinacosb = *sin(a + b)

13、 + sin(a-b)cos a cos b =扌cos(a + b) + cos(a-b) cosasinb = *sin(a + b)-sin(d-b)6. 萬能公式2 tan 2sin a =, >al+tan 21 - tail"2 cos a =1 + tan2 22 tan 7tana =.> al-tan 27 平方關(guān)系siii2x+cos2x = lsec2 x-tarr x = iesc2 x-cot2 x = l8. 倒數(shù)關(guān)系 taiixcotx = l9. 商數(shù)關(guān)系siiixtanx=cosxsecxcosx = lC5CX-sillX=lcosx

14、cot x =sinx十六、幾種常見的微分方程1. 可分離變量的微分方程：牛= /(x)g(y) ,(x)gx(y)dx+f2x)g2(y)tfy = 02. 齊次微分方程：=axI x 丿3. 階線性非齊次微分方程：字+p(x)y = 0(x)解為： ax尸汁皿膽皿dx+c三角函數(shù)公式兩角和公式sin(A+B) = sinAcosB+cosAsinBsin(A-B) = sinAcosB-cosAsinBcos(A+B) = cosAcosB-sinAsinBcos(A-B) = cosAcosB+sinAsinBtan(A+B) = (tanA+tanB)/(l-tanAtanB)tan(

15、A-B) = (tanA-tanB)/(1+tanAtanB)cot(A+B) = (cotAcotBl)/(cotB+cotA) cot(A-B)二(cotAcotB+1)/(cotB-cotA) 倍角公式tan2A = 2tanA/(ltan2 A)Sin2A=2SinA*CosACos2A = Cos"2 A-Sin"2 A二2Cos"2 A1=12sin"2 A三倍角公式sin3A = 3sinA-4(sinA)"3; cos3A = 4(cosA) 3 3cosAtan3a = tan a tan(n/3+a) tan(n/3a)半角

16、公式sin(A/2) = J (1cosA)/2 cos(A/2) = J (1+cosA)/2tan(A/2) = V (1cosA)/(1+cosA) cot(A/2)二 V (1+cosA)/(lcosA) tan(A/2) = (1-cosA)/sinA=sinA/(1+cosA)和差化積sin(a)+sin(b) = 2s in(a+b)/2cos(ab)/2cos(a)cos(b)sin(a)-sin(b) = 2cos(a+b)/2s in(ab)/2 cos(a)+cos(b) 二 2cos (a+b)/2cos(a-b)/2 -2sin(a+b)/2sin(a-b)/2tan

17、A+tanB二sin(A+B)/cosAcosB 積化和差sin(a)sin(b) = -l/2*cos(a+b)-cos(ab) sin (a) cos (b) = 1/2* sin(a+b) +sin (a-b) 誘導(dǎo)公式sin(a) = sin (a)cos (a) = cos (a)cos ( n /2_a) = sin (a) sin (刃/2+a) = ccos(a)cos(b)二 1/2*cos(a+b)+cos(a-b)cos(a)sin(b) = 1/2*s in(a+b)-sin(ab)sin(n/2a) = cos (a)(a) cos(n/2+a)二-sin(a)sin

18、(n-a) = sin (a)cos (口-a) = cos (a)sin (兀+a) = -sin (a)cos(n +a) = -cos(a) tgA=tanA = sinA/cosA萬能公式sin(a)二2tan(a/2)/(l+tan(a/2) "2cos(a)=1-tan(a/2)"2/l+tan(a/2) 2)tan (a) = 2tan(a/2) / l-tan (a/2) 2其它公式a sin (a) +b*cos (a) = V (a"2+b2) *sin(a+c)其中，tan (c) =b/aa*sin(a)-b*cos (a) = V (a2

19、+b"2) *cos (ac)其中，tan(c)=a/b 1+sin(a) = sin(a/2)+cos(a/2)J 2; lsin(a) = sin(a/2)-cos(a/2)"2; 其他非重點三角函數(shù)esc(a) = l/sin(a) sec(a) = 1/cos(a)雙曲函數(shù)sinh(a) = e a-e" (_a)/2 cosh(a) = e"a+e"(-a)/2 tg h(a) = sin h(a)/cos h(a)公式一：設(shè)a為任意角，終邊相同的角的同一三角函數(shù)的值相等: sin (2k n + a )二 sin a cos (2k

20、 n + a ) = cos atan (2k n + a ) = tan acot (2k 刃 + a ) = cot a公式二：設(shè)a為任意角，n + a的三角函數(shù)值與a的三角函數(shù)值之間的關(guān)系：sin ( Ji + a )二 sin acos ( Ji + a ) = -cos a tan ( Ji + a ) = tan acot公式三：任意角a與-a的三角函數(shù)值之間的關(guān)系：sin (a ) = -sinacos (一 a )二 cos a tan (- a ) = 一tan acot (-a )二 cot a公式四：利用公式二和公式三可以得到n-a與a的三角函數(shù)值之間的關(guān)系: sin (

21、 n - a ) = sina公式五：利用公式-和公式三可以得到2n-a與a的三角函數(shù)值之間的關(guān)系：sin (2 n - a ) = -sin a cos (2 - a ) = cos atan (2 n - a ) = tan acot (2 n - a )二 _cot a公式六：H /2± a 及3 n /2± a 與 a的三角函數(shù)值之間的關(guān)系：sin ( n/2+ a ) = cos acos ( 口/2+a ) = -sin atan ( n /2+ a ) = cot acot(n /2+a ) = -tanasin ( n /2 a ) = cos acos ( n /2- a ) = sin atan ( n /2-