数学(一)考研辅导高等数学讲义.pdf

1 2 3 4 5 6 7 8 9 10 11 12 1 1 1 1 Df xD yf D y f x x y D Z 2 2 11 xx yf xxx xx x 1 x 1 y f x 3 y f x F x y 0 y y x 4 y f x xy f x 1 xfy 1 yfx 1 y C 2 yx 3 x ya a 0 a 1 x ye e 2 7182 4 logayx a 0 a 1 10 loglgyxx yy f xx D 13 logln e yxx 5 sin cos tan yx yx yx cot sec csc yx yx yx 6 arcsin cos yx yarcx arctan cot yx yarcx lim arctan x x lim arctan x x 1 0 lim x x e 1 0 lim x x e 0 lim ln x x arctanyx x ye lnyx 1 yf u U ug x X U UU yf g x X u 2 1 y f x X M xX f xM f x X X xX fxf x f x X xX fxf x f x X y f x X 1212 xXxXxx 12 f xf x 12 f xf x f x X 1212 xXxXxx 1212 f xf xf xf x f x X 14 f x X 0T xX xTX f xTf x f x T f x 2 lnlnln100f xxx lnlnln x xe 2 100 x 2 10010 xx f x 10e 1 ln5 yxx x xx 1x 0 x 1 ln5x 546xxx 0144 55 66 f x 0aaa 2 1f x 2 1axa 2 11axa 1a 10a 2 1xa 1xa 01a 10a 11axa 11axa 11axa 102 2 24 x g x x 21f xgxg x 3 2 f g x 0 4 024x 02x 014x 15x f x 2 31 32 13 22 fgg 15 1 33 1x ye 3 3 33 11 ln 1 ln 1 yx y x 3 3 1 1 ln x y 0y 1y 0 1 1 1 f x g x f g x 1 1 x f x x 1 1 f f x 1 11 11 x f x xx 1 1 1 x f x 1x 111 1 1 1 12 xx ff x f xxx 1 2xx 2 2 1 x f x x n fff xfx n 2 2 2 222 1 1 11 12 f xxxx fxff x x xfxx 2 1 k x fx kx 2 1 2 22 1 1 11 k k k fxxx fx kx kxfx 2 1 1 x kx n 2 1 n x fx nx 16 2 g x f g x f x 1 2 1 xxx f eeex f x 1 x eu ln 1 xu 22 1 1 ln 1 ln 1 f uuuuuuu 2 ln 1 f xxxx 2 xx f exe 1 0f f x ln x et xt ln x t f ef t t 22 1 1 ln11 1 lnln 22 x xt f xfdttx t 1 0f 2 1 ln 2 f xx 1 F xf x A f x F x B f x F x C f x F x D f x F x B 3 2 1 3 x f xxF x C cos1 sinf xxF xxx D 2 2 f xx F xx A 0 0 x F xFf t dt f 00 0 0 xx FxFf t dtFfu du 0 0 x Ff u duF x F x 2 1 52 1 ln 1 xx Ix xeexxdx 1 xx f xee 2 112 ln 1 xx fxeef xfxxx 17 3 22 2 2 2 1 ln 1 ln 1 xx fxxx xx 2 2 ln1 ln 1 xxfx 2 ln 1 xx x eexx 11 66 10 2 02 7 Ix dxx dx f x g x 0fx g xf x g x axb A f x g bf b g x B f x g af a g x C f x g xf b g b D f x g xf a g a 2 1 0 f x fx g xf x g x g xgx f x g x xN n xA 2 lim x f xA 0 X x X f xA 3 lim x f xA 0 X x X f xA 4 lim x f xA 0 X x X f xA 0 lim xx f xA 0 0 0 xx f xA 0 lim xx f xA 0 0f x 0 0 0 xx f xA 0 lim xx f xA 0 0f x 0 0 0 xx f xA 0 0f x f x 0 x 0 0f x f x 0 x lim fxA n xf n 19 lim fxA lim f xB AB lim fxA limg xB x f xg x AB AB x f xg x 00g xB lim f xA x f x lim fxA limg xB lim f xg xAB lim f xg xAB lim f x g xA B lim f xA g xB 0B lim g x B f xA 0A lim0fx f x x 1 lim0 xx x 1 x 0 xx 1 x 0M x f xM f x lim fx x f x 1 fx f x 0f x 1 f x 20 lim f xAf xAa x lim0a x lim0 lim0f xg x lim f x l g x 0l f x g x f xo g x g x f x 0l f x g x 1l f x g x f xg x 0 x sintanarcsinarctanxxxxxxxx 2 1 1 cos1ln 111 2 a x xxexxxxax a 1nn xx n n xm n lim n n xA Am 1nn xx n n xm n lim n n xA Am g xf xh x limlimg xAh xA lim fxA 0 sin lim1 x x x 1 lim 1 n n e n 1 lim 1 u u e u 1 0 lim 1 v v ve 21 0 x 2 1 2 n xn xx exo x n 3521 21 sin1 3 5 21 n n n xxx xxo x n 242 2 cos11 2 4 2 n n n xxx xo x n 23 1 ln 11 23 n n n xxx xxo x n 3521 21 arctan1 3521 n n n xxx xxo x n 2 111 11 2 nn n xxxxo x n 0 0 lim0f x lim0g x x fx gx lim fx A gx lim f x A g x lim fx gx lim f x g x lim f x limg x x fx gx lim fx A gx lim f x A g x 22 00 0 0 lim x f xxf x fx x 1 1 1 lim 0 n n k k ff x dx nn n 0 b0 m a 1 110 1 110 lim mm mm nn x nn a xaxa xa b xbxb xb 1 110 1 110 lim mm mm nn x nn a xaxa xa b xbxb xb 11 110 11 110 lim m nmm mm nn x nn xaaxa xa x bbxb xb x 0 m n mn a mn b mn 2 0a 1r 1 lim n n aarar 1 1 lim lim 11 n n nn ra aarara rr 23 1 2222 lim 1 3333 n n n 2 3 a 2 3 r 2 2 3 25 1 3 11 1 2422 lim 3 3 11 1 2 33 n n n 23 3 1 1 32 lim 23 nn nn n 2 3 3 lim3 2 21 3 n n n 1r lim0 n n r 4 l 1 1 lim n n k k kl 11 11 k kllkkl 1 111111 1 21 n k k klllnnl 111 1 2ll 1 1 lim1 1 n n k k k l 1 1113 lim1 2 224 n n k k k l 6 d 0 222 111 1 lim n dnd nnn 2 1 lim11 1 22 n nd nd n 1 d 222 121 lim 2 n n nnn 2 d 222 1321 lim1 n n nnn 24 7 0 11 lim x xx x 33 0 11 lim x xx x 0 112 lim1 2 11 x xx xxx 0 1111 lim x xx x 0 1 22 lim1 x x x x 0 11 2 12 1 lim1 1 x xx 22 0 3333 112 lim 3 1111 x xx xxxxx 1 1 8 1 1r 22 lim 1 1 1 n n rrr 2 222 111 lim 111 23 n n 1 2 11 lim 11 n n r rr 111111 lim 111111 2233 n nn 1 3 2 41111 limlim 2 2 3 322 nn nnn nnn 25 x 0 2 sincoscoscos 2242 lim 2 sin 2 n nn n n n xxxx x 1 11 2coscoscossin 2422 lim 2 sin 2 n nn n n n xxxx x sinsinsin 2 limlim 2 sinsin 22 n nn n nn x xxx xx xx 2 lim1 sin 2 n n n x x 2 1 10 2 lim 1 x x x 1 0 1 lim 1 x x x x 1 2 10 10 2 22 lim 1lim 1 xx x x nx xx 10 2 1 2 2 2 lim1 x x x e x 11 1 1 1 2 00 1 0 0 lim 1lim 1 1 lim 1 lim 1 xx x xx x x x xx xe e xee x 12 11 21 2 000 1122 limlimlim 1 111 x xx xx xxx xxxx e xxx 3 1 cot 0 lim 1tan x x x 4 1 1 lim x x x 2 cot 0 lim cos x x x tan xt 1 cot x t 0 x 0t 1 cot 00 lim 1tan lim 1 x t xt xte 1 limcoscoscos 242n n xxx x 0 1 26 1xt 1xt 1x 0t 4 44 1 4 1 100 limlim 1 lim1 xt t xtt xtte 2 2 2 2 2 cos 1cos cot22 2sin2 sin 000 lim cos lim 1 sin lim 1 sin x x x x x xxx xxx 1 2 e 1 1 3 521 lim 2 4 62 n n n 1 3 521 2 4 62 n n x n 2 42 3 521 n n y n 0 xn yn 2 1 0 21 nnn xx y n 2 lim0 n x x 2 1 2 1 1 lim n n knk 2 1 lim n n k k nnk 1 222 1 1 1 n k nn nnnkn 2 1 limlim1 1 1 nn n nn n 2 2 1 limlim1 1 1 1 nn n n n 2 1 1 lim1 n n knk 222 1 1212 1 n k nkn nnnnnknn 2 1 1 121 2 limlim 2 2 2 nn n n n nnn n 22 1 1 121 2 limlim 112 nn n n n nnnn 2 1 1 lim 2 n n k k nnk 27 1 22 1 lim n n k n nk 22 222222 1 1 n k nnn nnnkn 2 22 1 lim 2 n n nn 2 22 lim1 1 n n n 222 11 11 limlim 1 nn nn kk n nkn k n 1 1 20 0 arctan 14 dx x x 2 1p 1 12 lim ppp p n n n 1 1 lim p n n k k nn 1 0 p x dx 1 1p 0 0 1 3 11 sin lim 1 sin n nn n 33 00 sinsin limlim sin xx xxxx xx 2 00 1 cossin1 limlim 366 xx xx xx 1 6 28 2 2 1 10 0 lim x x e x 0 0 2 2 1 1 3 912 00 2 limlim 105 x x xx e ex xx 2 1 t x 2 1 5 105 0 limlimlim t x t xtt eet xte 4 55 limlim0 tt tt t ee 1 0 11 lim 1 x xx e 00 11 1 limlim 1 1 x xx xx ex xex e 0 0 00 1 limlim 1 xx xxxxx xx ee exeeexe 0 11 lim 22 xx 2 2 22 0 1cos lim sin x x xx 222 22 0 sincos lim sin x xxx xx 22 4 0 1 sin 2 4 lim x xx x 3 0 4 2sin2 cos2 4 lim 4 x xxx x 3 0 1 sin4 4 lim 2 x xx x 2 00 1 cos44sin44 limlim 6123 xx xx xx 29 3 2 0 lim sinln x xx 2 2 00 ln limlnlim xx x xx x 1 3 0 lim0 2 x x x lim g x f x lim ln g xf x e lim ln g xf x 1 2 sin 0 lim x x x 2 sin x yx 2 lnsinlnyxx 2 00 lim lnlim sinln0 xx yxx 0 0 lim1 x ye 2 2 cot 0 lim cos x x x 2 cot cos x yx 2 lncotlncosyxx 2 22 0000 lncoslncos limlnlimcotlncoslimlim tan xxxx xx yxx xx 0 0 0 tan1 lim 22 x x x 1 2 0 lim x ye 3 11 lim sincos x x xx 11 sincos x y xx 11 lnln sincosyx xx 0 11 ln sincos ln sincos limlnlimlim 1 xxt ttxx y t x 0 cossin lim1 sincos t tt tt lim x ye 30 1 32 2 1 limsin1 31 n nn n n 3 32 23 111 1 limlim0 1 31 3 nn nn nnn n n 2 sin11n 2 0 1 cos2 arctan3 lim 1 ln 12 sin5 x x xx exx 2 0 1 2 3 3 2 lim 2 5 5 x xx xxx 3 2 0 1 3sincos lim 1cos ln 1 x xx x xx 0 0 0 sin1 3cos 13 lim ln 1 1cos2 x x x xx x x x 1 3 5 0 1 sin 6 lim x xxx x 35 5 sin 3 5 xx xxo x 0 x 5 5 5 0 11 5 lim 5 120 x x o x x 31 1 0 2fx 00 0 3 2 lim x f xxf xx x 0000 0 3 2 lim x f xxf xf xxf x x 0000 00 3 2 3lim2 lim 32 xx f xxf xf xxf x xx 000 3 2 5 10fxfxfx 2 yf x sinyx 2 lim n nf n 0 0f 0 0 sin 1 x fx 2 0 2 limlim22 0 2 2 0 nn ff n nff n n 1 0 a 1 0 xb 21 1 1 2 a xx x 1 1 1 2 nn n a xx x lim n n x 1 1 0 nn n a xxa x 2 1 1 0 22 n nnnn nn axa xxxx xx 1nn xx n x 1 lim n n xA 1 1 1 2 nn n a xx x 1 2 a AA A 2 Aa A 0 Aa lim n n xa 32 1 2 sin2 0 1 cos x x x f x x x x 00 sin2sin2 00 limlim 22 2 xx xx f xx 22 00 2 00 limlim2 1 1 cos 2 xx xx f x x 0 lim 2 x f x 2 1 4 0 2sin lim 1 x x x ex x e 1 4 0 2sin lim2 11 1 x x x ex x e 43 4 0 2sin lim0 11 1 xx x x eex x e 1 4 0 2sin lim1 1 x x x ex x e 1 2 2 1 lim3 sin 1 x xaxb x a b 2 1 lim 0 x xaxb 2 22 11 22 limlim3 sin 1 2 cos 1 2 xx xaxbxaa xxx 4 5ab 33 2 2 00 1 lim1 sin x x t dt bxxat a b 2 0 lim cos x xax bx 1b 1b 2 00 122 limlim 1 cos xx x xaxaxa 2 1 a a 1 3 0 x yf x 0 x x 0 x y 0 lim0 x y 00 0 lim0 x f xxf x yf x 0 x yf x 0 x yf x 0 x 0 xx f x 0 x 0 f x 0 0 lim xx f xf x yf x 0 x 00 0 limlim xxxx f xf xf x 00 0 limlim xxxx f xf xfx 0 x 0 x 34 ab yf x 0 0 lim xx f xf x f x 0 x 0 0 lim xx f xf x f x 0 x yf x 0 x f x 0 x yf x ab f x ab yf x a b f x ab yf x 0 x 0 x f x 0 x yf x f x 0 x 0 x f x 0 x sin x f x x x f x x 1 f x x 1 sinf x x I I I ab f x f x f x ab 35 f x ab M m M m 0 f xM ab 0 x ab x f xM M f x ab m f x ab M m m M C ab fC f x ab f a f b ab 0f 1 0 00 1 sin0 x ex f xx xx x 0 x 1 00 00limlim0 x xx ff xe 00 1 00limlimsin0 xx ff xx x 00f 00000fff f x 0 x 36 ln 1 0 1 0 2 11 0 x x x f xx x x x 0 x 1 00 ln 1 00limlim ln 1 1 x xx x fx x 00 1111 00limlim 211 xx x f xx 0000ff 0 lim x f x f x 0 x sin 0 0 x x f xx kx 0 x k 00 sin limlim1 xx x f x x 0fk 1k 1 sin0 0 1 sin0 xx x f xpx xqx x 0 x p q 00 1 limlimsin1 0 xx f xxfp x f x 0 x 1p 00 1 limlimsin 0 1 xx f xxqq f x f x 0 x 1q 37 3 1 1 x f x x 1x 33 1132 33 11 limlim 1 11 xx xx x xxx 3213 11 lim 3 1 x xx 1x f x 2 2 2 4 xx f x x x 0 x 2 2 0 x 2 2 0 x 0 2 1 00 lim 2 2 2 x x x f x xx 0 2 1 00 lim 2 2 2 x x x f x xx 0 x 2x 2 2 20 20 lim 2 2 x x x ff x xx 2x 2x 2 2 1 20 20 lim 2 2 4 x x x ff x xx 2x 1 2 4 f f x 2x f x lim x f xa 1 0 0 0 fx g xx x A 0 x g x B 0 x g x C 0 x g x D g x 0 x a 00 1 lim limlim xxt g xff ta x 0a 0 x g x 0a 0 x g x D 38 f x g x f x 0f x g x A g f x B 2 g x C f g x D g x f x A 1 0 1 0 x g x x 1f x 1g f x B g x 2 1g x C A f x g x 1f g x D g x h x f x g xh x f x g x f x 1 f x 0 x f x 00 0 lim lim xxxx f xfxf x 0 x 0 sin limarctan x x x 0 sin lim1 x x x arctanyu 1u 00 sinsin limarctanarctan limarctan1 4 xx xx xx f x x 2 3f 2 2 14 lim 24 x f x xx f x x 2 3f 2 lim 3 x f x 22 222 14 2 41 lim lim lim 2244 xxx x f xf xf x xxxx 22 13 lim lim 24 xx f x x 39 f x ab f aa f bb f xx ab g xf xx g x ab 0g af aa 0g bf bb g x ab f xx ab 4cos xx eex cos4 xx f xeex f x 0 0 30f 22 2 cos240fee f x 0 2 0 2 0f sin00 xx fxeexx f x 0 f x 0 f x 0 f x 40 yf x 0 x x 0 x x 00 yf xxf x 00 00 limlim xx f xxf xy xx f x 0 x 0 fx 0 0 0 x x x x df xdy y xx dxdx yf x 0 x yf x 0 x 0 xxx 0 xxx 0 0 0 0 lim xx f xf x fx xx 0 000 0 0 0 limlim xxx f xf xf xxf x fx xxx 0 000 0 0 0 limlim xxx f xf xf xxf x fx xxx f x 0 x f x 0 x yf x 0 x 0 fx 0 fx yf x 00 xf x 000 yf xfxxx 000 0 1 0yf xxxfx fx 41 S t Sf t 0 ft 0 ft 0 t yf x 0 x f x 0 x yf x 0 x 0 x yf xx 0 0 x yf x 0 x x 00 yf xxf x 0 0yA xxoxx 0 A x x ox 0 x x f x 0 x y 0 A xx f x 0 x 0 x xdy 0 x x df x dx x 00 yf xxf x yf x 0 x x 0 f x 0 x xdy yf x 000 Mxf x f x 0 x f x 0 x 0 00 x xdyA xxfxdx yf x dyfx dx dy fx dx 42 yf x yfx 0 x yfx 0 x yf x 0 x 0 x x y 0 yx 0 2 2 x x d y dx f x 0 x yf x 1n yf x n n nn n d y yfx dx yf x n 0c 0d c 1aa xaxa 1aa dxaxdxa sincosxx sincosdxxdx cossinxx cossindxxdx 2 tansecxx 2 tansecdxdx 2 cotcscxx 2 cotcscdxxdx secsec tanxxx secsec tandxxxdx csccsc cotxxx csccsc cotdxxxdx 1 log01 ln a xaa xa log01 ln a dx dxaa xa 1 lnx x 1 lndxdx x ln01 xx aaaaa ln01 xx daaadxaa xx ee xx dee dx 2 1 arcsin 1 x x 2 1 arcsin 1 dxdx x 2 1 arccos 1 x x 2 1 arccos 1 dxdx x 43 2 1 arctan 1 x x 2 1 arctan 1 dxdx x 22 22 1 ln xxa xa 22 22 1 lndxxadx xa 22 22 1 ln xxa xa 22 22 1 lndxxadx xa f xg xfxgx f x g xfx g xf x gx 2 0 f xfx g xf x gx g x g xgx df xg xdf xdg x df x g xg x df xf x dg x 2 0 f xg x df xf x dg x dg x g xgx yf uux x x f u u yfx x dydy du fxx dxdu dx dyfu dufxx dx dyfu du u xtyt yy xtt 0t 0 tdy t dxt 44 2 32 1 dydy dd ttttd y dxdx dx dxdxdt t dt yf x xg y 0fx 11 0gyfx fxfg y 1 d d gyfx gy dy dydx dx 33 0 fg yfx fx fxfg y yy x 0F xy y 0F xy x y y y 22 12200 x xyxy yyy y y g x yf x lng xf x ye 45 0 limlim xaxa f xf axa g x fa xaxa lim xa g xg a nn f xxaxa fa xa a 0 limlim nn xaxa xaxa f xf a fa xaxa 121 lim0 nnn xa xaxaxa 0 0 xx yf xx xx 0 0 x yf xx 0 0 x 00f 00 limlim0 xx f xfx 00 limlim0 xx f xx 0 lim00 x xf 0 0 x f x 00 00 0limlim xx xy f xx 00 limlim1 xx xx xx 00 00 0limlim xx xy f xx 00 limlim1 xx xx xx f xxa g x g x a fa g x 00ff yx 46 2 1 1 xx f x axbx ab f x 1x f x 1x 2 11 1 0limlim1 xx ff xx 11 1 0limlim xx ff xaxbab f x 1x 1ab 1ba 2 111 11 1limlimlim12 11 xxx f xfx fx xx 111 111 1limlimlim 111 xxx f xfa xaxb fa xxx f x 1x 11ff 2a 211 21aba f x 1x 22 1ln 1 yxxx 22 cotarccos 1yxx 2222 1ln11ln1yxxxxxx 22 222 1 ln111 111 xx xxx xxxx 2 2 ln11 1 x xx x 22 cotarccos 1yxx 2 22 1 2cot csc 1 11 x xx xx 3 2 2cos sin 1 xx x xx 47 2 sin x yex lncot sin xx y x 2222 cos sinsin2sin 2 xxxx x dyxdee dxxexdxe dx x 2 cos 2 sin 2 x x exxdx x 2 sin lncot lncot sin sin xdxxxx dx dy x 2 11 csccos lncot sin x dxxxx dx xx 3 1 csccos lncos cot sin xxxxx dx xx 1 2 100 f xx xxx 50 f 1 2 49 51 100 g xx xxxxx 50 f xxg x 50 fxg xxg x 502 50 50 50 1 50 50 fg f x ln f x yfx e dy ln ln f xf x dyfx deedfx 1 ln ln f xf x fx efx dxfx edx x 1 ln ln f x efx fxfxdx x 48 yy x 22 arctan x xyye dy dx dy x y x 2 22 1 22 2 1 xx y xyyy ee x xy 22 2222 22 2 11 xx yyx yee x xyxy 2 222 22 2 22 2 22 2 214 1 2 421 1 x x xx yx e xyexyxx xy y y ey xxxye xy 2 22 2 22 14 421 x x yexyxx dydx y xxxye 22 arctan x dxyd ye 22 2 22 1 1 xx d xyedyyde xy 2 22 2 2 1 xx y xdxydyedyedx x xy 22 2222 22 2 11 xx yyx edyedx x xyxy 22 2222 22 2222 42114 2121 xx y xxxyeyexyxx dydx xxyxxy 49 2 22 2 22 14 421 x x yexyxx dydx y xxxye 2 22 2 22 14 421 x x yexyxx dy dx y xxxye 3 2 1 2 1 x x xx y xex y 2 1 lnlnln 1 ln 2 ln 1 ln 3 x yxxxxex x 2 11 11121 3121 x x xe y yxxxxex 3 22 1 1 2 11121 312 1 1 x xx x xxxe y xxxxexxex 3 2 ln 1 sin xt ytt dy dx 2 2 3 2 sincos 3 1 dy dytttt dt dx dxt dt t 323 2 12 sincos12sincos 33 ttttttttt tt 1 0 yx x 0 0 1 x x 0 2 0 0 11 yxx xx 0y 0 2Xx 50 0 x 0 2 Y x 0 0 112 2 2 22 SXYx x 2 3 1xt yt 2t 2 0 125x 3 0 28y 2 33 22 dyt t dxt 2 3 t dy dx 83 5 yx 370 xy 1 22 lnyxxa y 22 22 1 yxxa xxa 222222 11 1 x xxaxaxa 3 22 2 3 22 1 2 2 x yxax xa 2 arctan ln 1 xt yt 2 2 d y dx 2 2 2 1 2 1 1 t dy dy tdt t dx dx dtt 2 2 2 2 2 2 1 1 1 dydy dd d ydxdxdx t dtdtdxdx t 51 yy x 22 1xy y 220 x yy x y y 2 22 1 x y yxyy y yy 22 33 1yx yy n n y y n y n x ye nx ye 2 0 1 x yaaa ln nxn yaa 3 sinyx sin 2 n n yx 4 cosyx cos 2 n n yx 5 lnyx 1 1 1 nnn ynx k yx k n y n 1 1 0 kn n k kknxnk y nk 1 n x y x n y n 12 11 1 1 11 n nn x yxxx xx 1 1 1 1 n n n n yx x 52 2 1 32 y xx n y n 11111 21 1221 yxx xxxx 22 2 1 yxx 33 1 2 2 1 yxx 1 1 1 2 1 nnnn ynxx 44 sincosyxx n y n 22 1cos21cos2 22 xx y 2 131 22cos 2cos4 444 xx 1 1 4 cos 44cos 4 422 nnn nn yxx 53 2 2 f x 1 a b 2 a b 3 f af b ab 0f 1 yf x A a f a B b f b A B 2 yf x A B x yf x yf x x f x ab ab ab f bf a f ba f bf afbaab 54 000 01f xxf xfxxx 0 x a b x yf x A af a Bf b yf x yf x f x ab 0fx f x ab f xg x ab fxgx ab f xg xc c f af b f x g x ab ab 0gx ab f bf af ab g bg ag g xx AB xg t tab yf t A g af a B g bf b AB AB 55 n f x 0 x n 2 000 0000 1 2 n n n fxfxfx f xf xxxxxxxRx n 0 xx 00 n n Rxoxxxx 0 0 lim0 n n xx Rx xx n sincosln 1 1 xa exxxx a n n f x 0 x ab 1n ab n xab 2 000 0000 1 2 n n n fxfxfx f xf xxxxxxxRx n 1 1 0 1 n n n f Rxxx n 0 x x 0 x n 0 0 x n lim0 n n Rx f x 0123fff 31f 0f 56 f x f x M m 0 mfM 1 mfM 2 mfM 1 0 1 2 3 mfffM c 1 0 1 2 1 3 f cfff 3f cf f x 0 3 0f f x 01 2 3 1 30f x dxf 0f 2 1 3 c 2 3 1 2 1 3 f x dxf c 2 3 1 3 0 f cf x dxf f x 0c 0 01 c 0f 0 x ln 1 1 x xx x ln 1 f tt 0 x 1 1 ft t 1 ln 1 ln10 1 xx 0 x ln 1 1 x x 0 x ln 1 1 x xx x 57 f x ab ab f af b ab 0f ca b f cf af b f cf a f x ac 1 a c 1 0 f cf a f ca f bf c f x cb 2 c b 2 0 f bf c f bc a b 0f 0fx 0 0f 1 0 x 2 0 x 1212 f xxf xf x 12 xx 1111 0 0 f xf xfxf 11 0 x 1221222 f xxf xxxxf 2212 xxx 12 0fx fx 12 ff 1122 f xf xxf x 1212 f xxf xf x f x ab ab 0ba a b a b 2 ab f f g x 58 ff bf afba g bg ag bg ag 22 2 fffba ab ba 2 g xx 22 2 3 fbaba f 3 g xx f x 01 0010fff 11f 4f f x 0 x 2 1 1 00 2 f xffxfx 1 0 x f x 1x 2 2 1 1111 2 f xffxfx 2 1 x 1 2 x 1 11 28 ff 1 1 0 2 2 11 1 28 ff 2 1 1 2 12 8ff 12 8ff 12 max 4ff 4f 59 f x ab 00fx f x ab 00fx f x ab f x a a 00fx 0f a xa 00f x f x ab 0 x ab 0 x 0 x xx 0 f xf x 0 f x f x 0 x f x 0 x 0 x xx 0 f xf x 0 f x f x 0 x f x f x 0 x 0 x f x 0 0fx 0 0fx 0 x f x f x 0 x 0 0 xx 0 fx 0 0fx 1 00 xx x 0fx 00 xx x 0fx 0 f x 0 x 2 00 xx x 0fx 00 xx x 0fx 0 f x 0 x 3 00 xx 00 xx x fx 0 f x 0 x 60 f x 0 x 00 00fxfx 0 0fx 0 f x 0 x 0 0fx 0 f x 0 x f x ab f x ab 1 x k x 1 f x k f x f a f b 1 f x k f x f a f b f x ab M f x ab f x I 1 x 2 x 1212 1212 11 2222 xxxx ff xf xff xf x f x I yf x yf x yf x yf x f x ab fx ab x 0fx yf x ab ab x 0fx yf x ab 61 yf x fx 1 x 2 x k x lim xa f x lim xa f x xa yf x lim x f xb lim x f xb yb yf x lim0 x f x a x lim x f xaxb lim0 x f x a x lim x f xaxb yaxb yf x yf x fx 0fx fx fx fx fx fx yf x 62 yf x M x y 3 2 2 1 y k y 0k 1 R k M x y M D MDR D D R 0 1 0fx 0f 1f 10ff 01ff A 1010ffff B 1100ffff C 1010ffff D 1010ffff B 101 0ffff 01 0fx fx 1 0fff f x a b a b 0f a fx f x x xa a b 2 xa fxf x x xa f xf xf afxa ax fxf x xa fx fx f 0 x x a b 63 f x f x A B C D max 2 1f xxx 1 1 1 3 1 2 1 3 xx f x xxx 12 33 f f x 0 x 0 0 0 lim n xx f xf x k xx n 0k n 0 f x 0 0 n f xf x kx xx 0 lim 0 xx x 000 nn f xf xk xxx xx n 0 xx 0 f xf x k k 0 0 f x k 0 0 f x n 0 xx 0 f xf x 0 x 0 f x 64 2 3 2 1 3 f xxx 3 1 3 3 21 1 22 1 3331 x fxx x 0fx 2x 1x fx 1x 0fx 12x 0fx f x 1x 2x 0fx f x 2x 2 1 3 f 1 2 3 f 2 2 1 1 x y x yf x 1 1 3 2 1 1 x fx x 4 4 2 1 x fx x 0fx 1 1x 0fx 2 2x 2 2x 1 1x f x 2 2 1 1 1 1 fx fx x fx fx y f x y x 5 2 9 f 1 1 2 f 5 3 2 17 4 9 65 f x F x I Fxf x I F x f x I f x I f x I f x dx x f x f x dx f x dxF xC F x f x C Fx dxF xC dF xF xC f x dxf x df x dxf x dx kf x dxkf x dx f xg x dxf x dxg x dx f x I f x I 2 sin x dx 2 cos x dx sin xdx x cosxdx x ln dx x 2 x edx 1 1 a a x x dxC a 1a 1 lndxxC x 1 ln xx a dxaC a 0a 1a xx e dxeC cossinxdxxC 66 sincosxdxxC 2 2 1 sectan cos xdxdxxC x 2 2 1 csccot sin xdxdxxC x tan secsecxxdxxC cotccscxcs xdxxC tanln cosxdxxC cotln sinxdxxC secln sectanxdxxxC clnccotcs xdxcs xxC 22 arcsin dxx C a ax 0a 22 1 arctan dxx C axaa 0a 22 1 ln 2 dxax C axaax 0a 22 22 ln dx xxaC xa 0a f u duF uC x ux fxx dxfxdxf u du F uCFxC 1 f axb dxf axb d axb a 0a 1 1 nnnn f axb xdxf axb d axb na 00an 67 lnlnln dx fxfx dx x 2 111dx ffd xxxx 2 dx fxfx dx x 1 ln xxxx f a a dxf ad a a 01aa xxxx f e e dxf ed e sincosssinfxxdxfinx dx cosscoscosfxinxdxfx dx 2 tansectantanfxxdxfx dx 2 cotcsccotcotfxxdxfx dx secstansecsecfxecxxdxfx dx csccsc cotcsccscfxxxdxfx dx 2 arcsin arcsinarcsin 1 fx dxfx dx x 2 arccos arccosarccos 1 fx dxfx dx x 2 arctan arctanarctan 1 fx dxfx dx x 2 arccot arccotarccot 1 fx dxfx dx x 2 1 arctan 11 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