微积分公式毕业论文

1、考无忧论坛-考霸整理版a0b0（系数不为0的情况）有关高等数学计算过程中所涉及到的数学公式（集锦）n , n A ,.a0X + a/ +anlim mb0xb1xm+bm一 丰亦八土 /八” sin X 一、重要公式（1） lim=1(2) lim(1 + x)(3) lim 脳(a A 0)=1(4) lim 折=1n_lim兀arcta n x =2(6)兀lim arc tan x = 一J产2(7) lim arccot x =0Xf(8) lim arc cot x =兀(9)Xlim e =0X产(10) lim ex =处x_jioC(11) limX =1三、下列常用等价无穷

2、小关系XT 0)sin X : Xtan X : Xa r cs ixn： xarcta nx : xcox：-x22ln (1+x)：xe-1aX1 ： xl(1+xf-1:ex四、导数的四则运算法则(U v ) = U v(uv ) = u v + uv五、基本导数公式(sin X ) =cosx(cosx ) = -sin X2(tanx ) =sec x2(cot X ) = -csc X(7) (secx j =secx dan x(8) (esex ) = -cscx cot X(a =ax ln a 1(11)(ln X )=-X(12)(logaX )=X l n a， 1(1

3、3) (arcsin x ) =J1 X2 1(14) (arccosx) =- pJ1 -x2, 1(15) (arctan x )=1 +x2I1(16) (arccot X ) = 2()(1 +xX ) = 1 (18) ( JX ) = -4=2Jx六、高阶导数的运算法则(2)cu(x)F)=cu(n)(x )1） u（X ）v（x）F）= u（X ）v（x y）(3) u (ax+b ) 丁)= anuC *ax+b )(4) u(x)v(x)=送 cfW * x y(k) ( x)k)七、基本初等函数的n阶导数公式(3)(aX)()=axlnna(1) (xnf)= n!(2)

4、(eax* f =an eax枕2JL-/M n fJI(4) sin(ax+b)L = asin ax+b+n”一cos(ax+b)=an cos ax+b+ n、八、n In a心(T)卄(ax +b )微分公式与微分运算法则 d(xb= Pxdxlax +b 丿in(ax+ b)F )=(1 厂 a W j)!n(ax + b) d (C )=0 d(sin x)=cosxdx d (cosx )= -sinxdx d (tanx psec xdx2(6)d(cotx) = CSC xdx d (secx )=secx tanxdx d(cscx )= -cscx cot xdx d (

5、e=exdxxxd(a ) = a In adx1(12) d (logax ) =dxxln a1 1(13) d (arcs indx (14) d (arccos x )= - -彳? dxj1-x2j1-x21(15) d (arctan x ) =21 +xdx1(16) d (arc cot x ) = 2 dx1 + x九、微分运算法则 d (u v )=du dv d(cu)=cdu d (uv )=vdu +udv八、fu ) vdu -udv dU；v2十、基本积分公式 Jkdx =kx +c Jxdx二釘+cdx f一 = Inix +c xfexd =ex +c(6)

6、Jcosxdx=sinx + cJsin xdx = -cosx +c1 2f2 = fcsc xdx = -cot X + c sin X 、1 2 f 2 dx = fsec xdx = tan x + c 、cos x1(10) f2 dx = arcta n x + c1+x2(11) 1f dx = arcs in x + c卜一、下列常用凑微分公式积分型换元公式1ff (ax +b dx = 1 f (ax +b d (ax + b ) a u = ax + bJf(xHd(xH)u = x1ff(l nx) dx= ff(l nxd(l nx)、Xu = l n Xf f (ex

7、 ) exd = f f (ex p (ex )Xu =eJf(aX)aXdx= 1 Jf(axd(ax) ln aXu = af f (sin X ) cosxdx = f f (sin x p (sin x )u = sin Xf f (cosx ) sin xdx = - J f (cosx d (cosx )u = cosx2/f (tan X ) sec xdx = J f (tan x d (tan x )u = ta n XJ f (cot X ) csc xdx = J f (cot X d ( cot X )u = cot X1f f (arctan x ) dx f f (

8、arcta n x d (arcta n x )、1 +xu = arcta n x1f f (arcsin x dx - f f (arcsin x d (arcsin x )d-X2u = arcs in x十二、补充下面几个积分公式ftan xdx = Tn cosx +cJcotxdx =ln sin X + cJsecxdx =ln secx + tanx +c f CSCxdx = In |cscx - cot x + cX -a十X + a21 2d-arcta n-+ca +x a ava dx-arcsin+c2 2-x十三、分部积分法公式形如.n ax . A fx e d

9、x，令 un=Xdv = eaxdx形如fxn sin xdx 令 un=Xdv =sin xdx形如f xn cos xdx 令 u = xn，dv = cos xdx形如形如形如十四、 Jx2 a2fxn arctan xdx，令 u = arctan x，dv = xndxfxn ln xdx，令 u = In X， dv = xndxdx = In x + 4xfeax sin xdx， feax cosxdx 令 u= eax,sin x,cosx 均可。第二换元积分法中的三角换元公式(1) Ja2 -x2X = as i nt (2) Ja2 +x2X = at a nt (3)

10、a【特殊角的三角函数值】(1)sin 0 = 0兀1sin=-( 3)6一.兀73sin =32兀sin =1)2（1）cos0 =1(2)cos6731cos 一 =-32兀 cos2=0)(1)tan 0 = 0兀tan 6JItan=73 ( 4)tan3JI(1)cot 0不存在兀cot 十五、三角函数公式1.两角和公式sin (A + B) =s in AcosB +cosAs in Bsi nA- B )cos(A + B) =cos AcosB si n As in Bc o sZA- B )a2 +cx = asect不存在siAicio-sc oAs cos(5)sin 兀=

11、0(5)COS = -1(5)tan 兀=0（5）cot兀不存Ao s BsAin Bstan A + tan Btan (A +B)=1 -tan Atan Bc、 cot A cot B -1cot( A + B)=cot B +cot Atan A tan Btan (A - B)=1 + tan Atan Bc、 cotAcotB+1cot( A -B)=cot B - cot A2. 二倍角公式sin 2A = 2sin Acos A2 2 2 2cos2A = cos Asin A=12sin A = 2cos A1tan2 2tan2A1 -tan A3. 半角公式A fl co

12、sA sin =2A J1 +cosA cos = J2 V 2_ jh -cosAV1 +cosA sin A1+cosAco厶戶泌2 V1-COSA sin A1 -cosA4. 和差化积公式a +bsin a +sin b =2sin2 24 c a+ba-bcosa + cosb =2cos cos2 2COSa -b.ca+b . absina-Sinb=2cos sin2cosa - cosb = -2sin22a b sin2sin (a +b )tan a +tan b =cosa cosb5. 积化和差公式1 Lsin a sin b = 2 |_cos(a + b )cos

13、(a b1Lcosacosb = 2 Lcos(a + b )+cos(a - b)Sira ccb- siab ( s an )bco as ibT=-2 Lsian b - (san)b6.万能公式2taa2ta nasin a =21 +tan2 旦2.,2a1 -tan -cosa =2-1 +tan2 -2tana =彳.2 a1 - tan-27.平方关系2+2 .Sin X +cos X =12x2.sec X -ta n x =12 .2 csc X cot x = 1考无忧论坛-考霸整理版secx cosx =1cscx sin X =18. 倒数关系tanx cot X =

14、19. 商数关系丄sin xtanx =cosx丄 cosx cot x =sin x(X(X(X(X(X(X六、几种常见的微分方程1.可分离变量的微分方程dyfi(x )gi(y )dx+ f2(x )g2(y )dy = 02.齐次微分方程：史=dx3.阶线性非齐次微分方程dy + p(x)y=Q(x)解为:dx= e三角函数公式两角和公式sin( A+B) = sin AcosB+cosAs inB cos(A+B) = cosAcosB-s inAsinB tan (A+B) = (ta nA+ta nB)/(1-ta nAta nB) cot(A+B) = (cotAcotB-1)/

15、(cotB+cotA) 倍角公式sin( A-B) = si nAcosB-cosAsi nB cos(A-B) = cosAcosB+s inAsinBta n( A-B) = (ta nA-ta nB)/(1+ta nAta nB) cot(A-B) = (cotAcotB+1)/(cotB-cotA)tan2A = 2ta nA1-ta 门人2 A)Si n2A=2Si nA?CosACos2A = Cos2 A-Si 门人2 A=2Cos2 A 1=1 2si 门人2 A三倍角公式cos3A = 4(cosA)3 -3cosA n /3+a)a)tan( n /3sin3A = 3s

16、in A-4(si nA)A3;tan3a = tan a ? tan(半角公式sin( A/2) =-cosA)/2cos(A/2) = V (1+cosA)/2tan( A/2) =dc1sA)/(1+cosA)cot(A/2) =V (1+cosA)/oi3A)tan( A/2) = (1-cosA)/si nA=si nA/(1+cosA)和差化积sin (a)+s in (b) = 2si n (a+b)/2cos(a-b)/2 cos(a)+cos(b) = 2cos(a+b)/2cos(a-b)/2 tan A+ta nB=si n(A+B)/cosAcosB 积化和差sin(

17、a)-si n(b) = 2cos(a+b)/2si n (a-b)/2 cos(a)-cos(b) = -2s in (a+b)/2si n (a-b)/2sin (a)si n(b) = -1/2*cos(a+b)-cos(a-b) sin (a)cos(b) = 1/2*si n(a+b)+si n(a-b) 诱导公式cos(a)cos(b) = 1/2*cos(a+b)+cos(a-b) cos(a)si n(b) = 1/2*si n(a+b)-si n(a-b)sin(-a) = -si n(a) cos( n-Ot) = sin(a)cos(-a) = cos(a) sin( n

18、-a? = cos(a) sin( n /2+a) = cos(a)cos( n /2+a) Fin(a)sin( -a) = sin(a) cos( n +a) -cos(a)万能公式cos( -a) = -cos(a) sin( n +a)-=in(a) tgA=ta nA = sin A/cosAsin(a) = 2ta n( a/2) / 1+ta n(a/2)F2 tan(a) = 2ta n( a/2)/1-ta n( a/2)F2 其它公式cos(a) = 1-ta n(a/2)F2 / 1+ta n( a/2)F2a?si n(a)+b?cos(a) = V (a2+b2)*s

19、in其中c)ta n(c)=b/aa?si n( a)-b?cos(a) = V (a2+b2)*cos(|a其中，tan (c)=a/b1+si n(a) = si n(a/2)+cos(a/2)F2;1-si n(a) = si n(a/2)-cos(a/2)F2;其他非重点三角函数csc(a) = 1/s in(a)sec(a) = 1/cos(a)双曲函数sin h(a) = ea-e(-a)/2 cosh(a) = ea+e(-a)/2tg h(a) = sin h(a)/cos h(a)公式一：设a为任意角，终边相同的角的同一三角函数的值相等:sin ( 2k 7+ a) = si

20、n acos (2k n+ a) = cos atan (2k n+ a) = tan acot (2k n+ a) = cot a公式二：设a为任意角，n + a勺三角函数值与a的三角函数值之间的关系:sin (n+a ) = - sin a cos ( n+ a) = -COS a tan (n+a = tan a cot ( n+ a) = cot公式三：任意角a与-a的三角函数值之间的关系：cos (- a) = cos a tan (- a = - tan acot ( - a) = -cot acot ( n a) = - cot asin (-a = -s in a公式四：利用公

21、式二和公式三可以得到n a与a的三角函数值之间的关系:sin (n a) = sin a cos ( n- a) = -COS a tan ( n a) = - tan a公式五：利用公式-和公式三可以得到2n-a与a的三角函数值之间的关系:sin (2 n a) = -sin a cos (2 n a) = COS a tan (2 n a) = - tan 公式六：(Xcot(2 n- a) = -cot an /2 及3 n /2 与 sin ( n /2+) a cos sin ( n 120)= cos sin( 3 n /2+ )(= - cos(3 n /2 a) = -cos a 求导公式a的三角函数值之间的关系：cos ( n /2+) a -sin acos ( n /2 a) = sin acos( 3 n /2+)= sin a(Xtan ( n /2+ )= -cot tan ( n /2a) = cot tan(3 n /2+