2.2 CA´ LCULO DE DERIVADAS 51dx = 1 dy P
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d 0 z | ul x u | ` '©8 l 0h I© rrrr f(x) − f(a) rrrr > 1 x − a 2 fV (a) bIl |) =y fg| y( fa)l )xvs )lx%lxYxy© 7Pz¤ D fBD©l− xYδY1z7y2|(l (ya|) )w lz= fxl−l 7 1=W Bδaδ>s(Ba§ρ0)(xb=lS ρ)y B >l©l δ|x 10l(} sa| y I¦§8)lg) ¯ ¤ y wglBδ2(a)Q % V P xBl δ B(faρY −()b1x ) 0 y ) | Py¦g| w l|
7y l f7Q −2 x1l(e| y) ))yêP By ρ©(7 b}) Q rrrr f−1(y) − f−1(b) − f V 1 rrrr = rrrr x−a − f V 1 rrrr = rrrrr 1 − f V 1 rrrrr y − b (a) f(x) − f(a) (a) (a) f(x)−f(a) rrrr (a) rrrr x−a ε f V (a) 2 = rrr 1 rrr f(x) − f(a) − fV < 2 2 = ε. x − a f V (a) 2 f(x)−f(a) f V (a) x−a PN1¤¢©ãw.w r—oãl©lotpasl is|!e
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52 DERIVADASsf(x) = −x T x P 5 g(x) = x x l | j 1 m z t|xx 0xx −><h 00w.,)l Yx y lhy l | l ¤ wl Ss w ©l0x q w x )l x g+V (0) = 0 l 7 | l0¨ xyl (g Ð s g| 0 f)+V (0) x l | x j x xg f(x)m = g(−x) = j 1 m x > 0, x x −< 0 . 0 l 2. (Un ejercicio).— f(x) = 4x(1−x) 5e 8 }) } x w | y x | l s x l | F(x) j F V (x) = 0 = f p f f(x)myy l l©|x l k5l V | 7y l x 0j4%f 2−V (2 x8))xdm¦=q lf
'0V jscf)(z xw|f)¤2vm© ( xfx)V(=xw )0 fs jz wf| (l©xxu ) =m F V h¤ 4 Dl wu l 8 tuw ) | l |el }xlx dF y(Vx)l)=x l©fx V 8 | V (x) = 0 ê• f V (x) = 0 1 ê ê • fV 2 j f(x)m =0 f(x) = 1 x = 1 p 2s · 2 4 2 = 0ê 1ê ê c · · Yx y } | y 28 X q • fV j f2(x)m f2(x) = f(x) = 1 p 2 s 2 x = 1 s 1 2s 2 4 2 4 | 7 w | y xv Rx l | y y q37 .leÛ 7 © ª
¦¢D) eg¦ E r ©livxDV zsa7§d anw nP –xr ´e uêswim l |ay ¦l ds e w u8n| p rfod§ ugctxoV:| R exgl aw |gd0e}| L©l ex ibx w neÛ i0z.l—| 7y l l l |
7yl l£ (fg)(n)(x) = n õ n f(k)(x)g(n−k)(x) , kö ql V 0u¤k w=l } |l0ly ufww(0¤|©u)0 = Vz7z |7 fê | lxll | y l | § j gy (0I ))= g = 1 4 dz w 4 | ©l x z x ¤ wl ¤ b X¼Éɺº©À)ºÐº ¼YФúÊÃPú½¼P½À(DÌ(üü1n¿½Á¼i)û)ÌDg û¼⇒Å º¼X)ÀÀ¡ PVÌ Á(nü ü º ½Ê À)ø ©l p7u `o j8´ }DnÒl ±0Vr 7w ²'´©·wqlE7©lç ªYx¦
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h yl | ) ê ¨ w n x l l + 1) s n @H n õ n f(k)g(n−k) V (fg)(n+1) = p (fg)(n) V = P(n) F n G q kö k=0 = n õn f(k)g(n−k+1) + f(k+1)g(n−k) kö k=0 n õn n+1 õ n = f(k)g(n−k+1) + f(l)g(n−l+1) kö l − 1ö k=0 l=1 n+1 õ n õn f(k)g(n−k+1) =+ kö k − 1ö k=0
2.2 CA´ LCULO DE DERIVADAS 53»¡ ËË Å ÅÌDË ¼¦Ç Å Ë DÌ ¼ Ç º Ë Ì Ë ngÊi¼ ú Ï n+1 õ n + 1 f(k)g(n−k+1) , = kö k=0 w l q¤ wl dj nw+kDl 1u mh j }u |gD dz wg x § hs l yl | lz | x le2x(ax2 + bx + c) ¤ j nkmj + j k−n1m = w % © ©l x ky ) =q 0g¦ l z t| kD = l ny q+I1) | n + 1 j −n1m = nn+1m = 0 D0x ueh w reilv|lDa|e7yy dvl)a
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l 8z }x w sx l w ©l l |E¤ ïj 3 ñ l s l 1. P ••FÉ u nl)) %cl e¦io´ynn|nl |=>2 pl 0of0 stVs(fexl Vnl(¿)xc=)i©l al x=nlw xy0dy ne|y− 7 1e'
xTx)pl2x}oxP n lu 5e|wny7l lt%eV e n}|te rw o0 . —z | s f(x) = | n wgx xn 8| l l©x ) 7 }} • ) s l % l©xw y x lx u w l l l 7
l w 4| 0' 0 l | 7y l n<0 2. 5 F+uns xclioy n l e| sl logar´ıtmica y exponencial.— ) f(x) = }| sR | l xP x fV (x) = {¤z }| (x + h) − }| x = {}z } | } |õ h 1/h p e1/x = 1. h 1+ = h→0 h→0 xö qx s z xvl | u l | l 8 s I©0 g(x) = } u a x = u a e | s gs x l z á (a x > 0) gV (x) = (} u a x(+l a)0 )V >=s=05 )u x1¦s a©l t(}x aguxc}a)¤ ¤ |Be=w
.l lx2 x u yl y l s l©5x l 7 5
) w¦gg| lq0 l z || s l lhy gl |lD(xul ) a©x s § e|%V f h0 h | x ul w| s ga©l Rxx(5 x = l} l | g j g−1(x)m = x } u(ax)V 1 ae = 1, ax wDl u (ax) V = }= au| }ax| xe x==b es a |lxÛx} || s a§Au.x I )|x%<©y y ¤ w ) s (ex) V = ex ©¿ gw ©l l x e| x s }| =0se}x|l (y− lx| )l§ y © sx f(x) s x =W 8%qV q¦g lz¨ | flDVu (x ) =l x1¤ >0 0 x }|j (−x)m V = 1 1. −x (−1) = x
54 DERIVADAS e| q | w l
¦h © g } ¤ © d0 z | l lDu l u© l e| s gw x 8| ©l Yx y l © ª ¦I s }|j V = g V (x) . g(x) g(x) mbx3.>P F05us nx cwbi8o´=¤n¤ wepb l o1 ù txe|n's §ciyxalll|ydlule|x ell |xyp o¨neD nV |tex alr b)i tr¤ar wio|g.0— z | j I©) w | l0¨ %0 | l |©y l xb x < 1j xbm V = eb ù x b = bxb−1 . |xl4} xRl© jw } x0 g¦ ll s q q¤ ¦P7 w ± ylz | l q xq ll©l|Yx |yy z l}e|gÛ }| | V l}|huI l)| l l|8 xwwl0¨ yl | l)l x xz |=W 0 0 k5f l 7d¦
D80| z | }| SIgs )xl y xRg| 0 xq = q x (xq) V q =⇒ (xq) V = qxq−1 . = xq x l wu x l x l a )x l | y s F5 uxnl cyio l n| elssw cy i} r}}c)Ú u8 |la r2 es q.— V z 7 4. |Aw 7x} l | q + ) = x l |f(ax)0 =x b | 0x x b xP {¤z x l x l (a b) + hx l→| 0x f V (x) = | (x + h) − |x = {¤z + 0 0 xh −1 x x ¤{z xl | h = 0 x x . ht I ) y El )lz Yx y q ©l xhw → y 08| h h→0 h lhx l©xYx( y=0()7y xuxc! xxw1))} 4g| VV−0 ¤===} ygw |2ll©0pp jx5xxx0lxl1uy 2||w7{}xz Úxlxxju sπ2Vq w=g| −VlV=xw1 ml0z+z lqy D7V7y}lu)x |%=22xx0xxlx −+x¤y20¦
x)xxwlx|8l|j2xÛπ2x| P−| ¤ 0y 0}Vx¡−e|m |x xl | x Tx P 5, =− . x l w g| 0 } | s%}xB| lq
yl x =W 0, x x= l©x l ©l x x lx ds x | l | 0 y π2l©,Dx s π k 2 © x l | x l | x()) =x l x⇒ () xl | y)V 0 x = 0 1x x = x=1 ( | y)V c 1 ⇒ 1 − y2 T y P (−1, 1) .j {z Úqt |w )z } u 8 l % | y7xl sy I
¦)V x| Py |g[0 ,l©πDx ]s sfx v 0 x x = fs x ¨l y l | l x l | x= c 1 − y2 y © % 0 x (0 x x) = x ⇒ ()d 0 x y) V x l | x) = 1 (−
2.2 CA´ LCULO DE DERIVADAS 55 ⇒ () d0 x y)V = − x l 1| x = −c 1 TyP (−1, 1) . 1 − y2 V w z y 2 s () 7y u x) V = 1 T x P 5 F V zu7n cw io nx es w |4 l ¤5 .x 1+ xl2 z wl |© l l 0 ¤ 0}` ¥ V u 0 h d%¦ © hiperbo´licas.— ¥ ¥( x)V = x TxP 5 , ¥ ¥( x)V = x TxP 5 , ¥ ¥ ¥( t u ¥( x) V = 1 2x T x P 5 , 2x = 1− x) V = · 1 TxP 5 , t u ¥ x2 + 1 ( x) V = · 1 T x > 1, t u ¥ x2 − 1 ( 1 Tx P (−1, 1) . x) V = 1 − x26 .l Dw |ger0 }iv|a©lcx io´0 n loga rxR´ıtx mu w ic l a| .7y—©l x ê ) cV ¦ 7y l | l w |e Dl ¨ ©l x z |2I) c l 7 ¦
j V y7u s x)úüû ÷ f(x) = ( x j %¦ = x2/3 1−x x l | xl s j ) sf(x) 1 + x2 4x 3x f(x) = xxxx lh¥ d l } x uw l | 7y l ê x l y2(x )x =l 7}
| l | y f(x) Q g| l)Dx s wj l©y (lx)©m 8V } =w f V (x) %s lh | £l )V $s7 §y j V xl )| ¢ x 7 VfgV (l©x ) =l©xvf( xl ) j yx (x )um y(x)m ©f (¿sx)ew x l w j Ejercicio.– 8 | 1
)ln7 Ûg©I u) ¤ l) d 0n z | 7PV n¦I(© x5)¤ =w l dn (x2 −x l P r polinomiosde Legendre dxT nx P 5 (1 − x2)PnV V (x) − 2x PnV (x) + n(n + 1)Pn(x) = 0 . l g(x) = (x2 − 1)n e| l 7¦
8 D z t| } u ) {z y ¤© g V (x) =l 7 x
¦2)2|n−x1 w =⇒ (1 − x2)g V (x) + 2nx g(x) = 0.k5l g| (x¦l)s |I
l Ú sxl d¦ y l | l (1 − x2)g V V (x) + 2(n − 1)x g V (x) + 2n g(x) = 0 .
56 DERIVADASá l 7
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l )l xDs'wex ) | h q Vz 7 w ¤ l d l g¦ |} Ú s (1 − x2)g(n+2)(x) − 2nx g(n+1)(x) − n(n − 1)g(n)(x) ¢ £+ 2(n − 1) x g(n+1)(x) + n g(n)(x) + 2n g(n)(x) = 0 ,¤ wglx l x g} ªÛe© 2 q q V7!7V wl©x y Bs § h ¤ w l g(n)(x) = Pn(x)2l )lll.lx3h l
w e5g8 ©l7 }}YxVV yYxEs'y xl! e2L¨| ul.|g0¤¥w0TVxe|1
¦ gs)E7 xwlO©lz|g|¦g}| x0R 8lw y|ze|lE|| lllMl)|lxYl|4y l}A¨xc7} V lw8 |0x|eD0y uVz wE||g7l©|xYlLy | l0l ¨ygy luV
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!0 ) ¦gz)|¨ll ¤ }0 }w | l2l e|©V8 · f(x) = 3 x ; f(8) = 2 ; f V (x) = · 1 ; 1 f V (8) = ; 3 x2 12 ¤f(8 + 0.012) f(8) + 0.012 f V (8) = 2.001 .l©|) l I7ld} xl©Ix w|4| w zx | wgg u )l l5 |4 )l lx |% y l0 |) szl x ¦
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h l § sS f V (a) l | y |g )l x P¨S δrp7`oo>p8´ 0oDn Òsy±0i)rVc 7i²'o´¤ ©´ ·nwq Ól.ç a l© w DD z |ù8© ¦ x w vs x f w l 0 E %V l hg} f V (a) = 0 s f V (a) > V ¨ 5 0 f(x) − f(a) >0 T xP (a − δ, a + δ) ;E§ l | y >V |efx(l©a−xD)s a x d xwl©Pu (a −| δ,©l axv)w s| x 0ll '¨ yz fl (2x) < f(a) Q l 7 sx x P (a, a + δ) sx l z af(x)
2.3 EL TEOREMA DEL VALOR MEDIO 57 TEl¤ |wxl jehl lP mxx (ql}lp−| Þy7ylyx1olll,l.|1¦
| l)8l ys fx8f−yVw|(wg(x(lDu −0)ul )1|=) ,y7=01©l x)}x)l−ß x¥x1qw¦ l0|§¨'11yf−+u+wVl |(xx)z0y Ûgs))x = ©x l{zll 1|¤ y ¦
x l 2 w g| 0 z | f(x) = ¥ x2(1 + x) 1−x Isw{lz |l©l%xY} fy2 Vw (x |0)y
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l =ld | 0 0−< l l |f (yy xl) )x s V R y 7 s x s x =W 0 f V (x) = ¦ x x 1+x + (1 + x) 1 (1 − x) 3 x 1−x 2 2 ¦¦= x + 1+x x s − 1−x (1 + x) 1 (1 − x) 3 1+x x>0 2 2 1−x x x < 0 −x (1 + x) 1 (1 − x) 3 2 2 08wlls Dl7z0¤¦%u¨0y¦
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)g¦ l f V (c) §4l x P [a, es l c y P V e| (al©,x b) (x) = 0 =0 f T xP (a, b) b] | V } | Vc|`wl (y7vl©cl P)} |¦
l=(z7y)f¨a)l} |}0¤,©¦
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58 DERIVADAS | 7} l R w u ) x l
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¦qlx n2|kk0 ny7(Ûg=llxzz−l|¨s )y87}f1w(y1xV,x(2%.x jl.Vl)−fy|.7}VYx}=,Vl1|yV |uxnx})y| ln2ly−n−|n }xYkxxx1y }(l|lxzl |y7I72lf8)(xu−5xfxu−l 11|lk),YxYxg|ny1y (}−)=−||xR1kyy1yss,1xx1)l,wl )lx Dx.Dl|d{z.u sw.l(¤f7lD,−x w(lufnn1sl5xV ),}−1
¦yxDl ©llf)8u l1xYxh¢y|ly,wl| 4l0w||y |0l y¤¨ll%xDwg|ds|glf .}−(y.©lkf8| .Dx1()ns(ss 2−%)lf0|(V1ny§)ll−w¦11=n|ll)QE Vl l7 ! t
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¦y)0+w+ hk ¦s F§ xs(ξξl, |yy 0 ) , § x0 + h k5l | lx ) l 's § ¤ w l x0Fy =W 0ηl |l)xYyl z l f(x0 + h) − f(x0) = k = − Fx(ξ, y0) ; hl y hl 7
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2.3 EL TEOREMA DEL VALOR MEDIO 65 j ¤{z f(x) = {¤z g(x) = s h ¦e l | • x→a 0 } x→ {¤z a • j = +∞ x→a g(x) | y g| l©xDs'y 8 ¦g lz | 0l ¨ xY7y l l q {¤z f(x) =l x→a g(x) ¨ po`8´ nDÒ0± rd ²B)´ ·q çR¿l¥ x x z l $ ) x j 50 | a P 5 § {¤z fV (x) = l P 5 l δs[a>I,)0x ]4y 's 8 §) ¤x lw l¦y sx l x s l y g| l©x rrrδgfs x²² ((x→xxl ))a)+−g g}l¤ rrr©V (< x)jε ε > 0
¦Q)l }7S 0l 7<Ûg)x) −| a < | ¥ v l| l
0 δ −a < }| 7y l | l < x f(x) − f(a) f(x) fV (c) g(x) gV (c) yg (g|x)©l −Dx s g(a) s l©xw y = = , c P (a, x) . á l | D 0<c−a<δ rrrr f(x) − f(a) − l rrrr = rrrr fV (c) − l rrrr < ε. g(x) − g(a) gV (c) Ejercicios.— ec)} ) R } x }{z yl)x ¤{z (x − 2)e2x + (x + 2)ex (a) (ex − 1)3 ; x→0 (b) ¤{z õ yux 1/x2 ; (c) x→ }{z 0 xö p 7y u x 11/ ù x π − ) x→+∞ 2 q . xë l 2| x3{d³D ´ x s sxly l |1. Contraejemplo de lStolz. © A x w |g0 }| ©l x f(x) = x + x g(x) = eá÷ Cø ù x f(x) {¤z gf2VV ((d xx¤ ))w {z =¤ wg0l , § | l0¨ xYy7l l }{z f(x) . x w |g0}| l©x l©x 2 §x→0 ∞x xg=(x0) w x→∞ j w 8| s l | y l|y d | l ∞ g V (x) xl |x xl | s x ly l| l 2. 7 l | 0 g(x) = x + x © 0 f(x) = x− {¤z f(x) = 1 , § | l0¨ Yx y7l l ¤{z fV (x) . x→∞ g(x) x→∞ g V (x)µ # # ¶f+(a) = f (a+) § {}z B3.δY (aP)ropS oxs→icaio´fnV (.x) f sl©lx | 0y |g| y }l©| x w l | ¿s l ª¦
)g¦ l!l | w| l | y V7| 4 )l w 0 =l S fV a l (a) =
66 DERIVADASl| ¨ 7p o8´ Dn iÒ 0± Vr 7 '² ´)·q ç H2 l©x 0¤ w | ly } | w l | sS ¤{z f(x) = f(a) l w©l l y g| )l x )g} ¤ © ) Xj e8 d }{z f y l l gÛ | l a ê x→a f V (a) V {}z f(x) − f(a) = }{z fV (x) = l . x→a x − a x→a 1FfE§l)YÒ ul´8|( l¤xcnnw7lw)e|gdµ |¨r=0plarI¨l©mor©lxhxYuqr|w7yaxelr©Dlz·nefd¬y¤²(2nt¤xntlRa¤²w)VRnlew{lzd l`o|2dtxlm|eB´l|l)A|lyµl7yvx´el©V7`cp7u¦t¦g±´ao0%q%xlzl)dxlcVh´¤)ª¨l©uo©s¦
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lp) o¦gs luiclio´| nw .| x 0 | I I g(x) = x j g V (x) + x5 g(v) 2g V y(l 0 )
m 8 −} ¨ 1©8e¦ 0 l (ξ) , ©l vx w `| 3 w | y y | l ¦%s ξl | y f|gl© xl©vxw |exly l | § l0l B¨ y l 2 x l07ª§
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b] b−a º¹õ a + b − (b − a)5 f(v)(ξ) , f(b) ¸− f(a) = f V (a) + f V (b) + 6 4fV | l P)ldx (e ¤afwn|S,w l¢bui|em¶P)8 pr'
w ssx|go 0xnlyl| kz)q|| l=5 xl }f08| ( y7x=slk)
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¦ξ)k}P s ss[}}j a |x, +l | k(b − a) =!x 1l b2 )f.y .) .f x ¤ !2w nl k 2n 2|nxy w ¦g|g}| (a, fo´mula b b−a f(x) dx = y0 + 4y1 + 2y2 + 4y3 + . . . + 2y2n−2 + 4y2n−1 + y2n a 6n¨ 7pV (o`0)´8Dn=Ò0± grV7(i²'−v´©·)q (ç(20bj 8)8−=0 an0 )4s¢|50f(i7v}) (ξl x )l .0gw u ) s 7 %V ¦IV) ¤z }dz 0}I }©l 7y l f l g x l y7l | z g(0) = | g V
2.3 EL TEOREMA DEL VALOR MEDIO 67r§§t gψy }) }(q)Hx))g¦| | x l l l¨x y l x l w |g0} | l©x w ¨ } ) l©x sφ(x) = g(x) − x j g V (x) + 2 g V (0)m =lz | | 3 x5 φ(0) = =0y7ls l l |x Q 3 φ V (0) φ V V (0) = 0 φ V V V (x) = − g(iv)(x) x y ψ ©l (x0)
l = )l xvψxVDl(u0w) = x ψj V V (¥ 0) =v | l 2 x ψV V V (x) = 60 x2 >0 I φ(x) = φ V (ξ1) = φ V V (ξ2) = φ V V V (ξ3) = − 1 g(iv)(ξ3) , ψl $s (¥xx w) ψ V (|lξ|l 17ys )l s ψ V V (ξ2) ψ V V V (ξ3) 180 ξ3% V | j l©x s l | yl¦s xl
8y l sξ1 P (0, x) ξ2 P ê(0, ξ1) ξ3 P (0, ξ2) ê |I)} 2 ¤ w l g(iv)(0) = 0 g(iv)(ξ3) = g(v)(ξ) , ξ30 | ξ3) s §`¥ l xyl 7 }e| ξj ¦%P ( 0l , j s l ξÛy |VP e| ( 0¨©l ,x Ix)© 0xy 8 ¥t¤ s § aw +2 bgm l¨l | P gwz l y7¡l©−x bxq−2 al ,b)−2Ia) k y Gj ( x )s =§hs Q lF|` (7y tl)w s |g=ID) fz u| t©G+ %F(V x5 )−0 2F(| −x x u)w x> S 0 G(x) = xj − x5 G(v)(ξ) . G V (x) + 2 G V (0)m 3w s 180V l | BI ) y ¤ ¤) s I ) x= b−a Sz P j 0, b−a m y 8 c¤ w lAXj xwexYy 7y w §¦l | § 8| 2 2 ¸ º¹f(b) − f(a) = b − a f V (a) + f V (b) + 4 f V õ a + b − (b − a)5 G(v)(z) . 6 2 ö 2880l 7 zG0l)z y (¤y v©8 ) ( z¤ )lw| l =y F(v)(z) + F(v)(−z) s A§ ©l Yx y7!l l©x w Q|
8} Vu} | y l 7 ©l } j ©l ¤ ) 7 y l 2§ l j c¥ I© 0 w e| l 7 ¦
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(lv8| )t y7(slzgl s)|}©l ¤F+yx© (fl2w8vF|)yl(()vù|ξ)1(z©)−l g=z )l)fx(w=wv)g| y(Fξ()v )0l ( ξ|l 1% )xξ);IxP w)g¦(ya} |, ybl )
¦j %¦8}f x ¤[a, w |e0 s z | sF(x) = x f(t)sRd§ t a x2] . . . [x2n−2 , b] b b−a f(x) dx = y0 + 4y1 + 2y2 + 4y3 + . . . + 2y2n−2 + 4y2n−1 + y2n a 6n (b − a)5 f(iv)(ξ1) + . . . + f(iv)(ξn) , − 2880 n4 n
68 DERIVADASw0 e| ©l| ξ} kl l Pª|
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j 8 }¥5V5 }|Iy7©l 0£ e y %V f(ξk) P (a, b) f(iv)(ξ1) + . . . + f(iv)(ξn) = f(iv)(ξ) . n2Ù l7.B|4 V ¯ x l©| ©x wLsl©|g§Yx¢A0'y}Ü V}¢B© | w Fzjl©¢©qxyOdä´4¨à%©¤IlR¢7ÜÃ7 }
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y8 x8} a¤ © §wg¦ l l y l § l0f¨ w Yxe| y7l x a § x I f(x) = f(a) + (x − a) f V (a) + (x − a)2 f V V (a) + (x − a)3 f V V V (a) + . . . 2 3! + (x − a)n f(n)(a) + (x − a)n+1 f(n+1)(ξ) . n! (n + 1)!dl}©lt5¨ |vx exDl7p7y {lzo`hl gs fg
¦r8´ r(a8nD}exÒds)0±s torV=[oa7an'² ´©,Tn·<qxnd–ç]fe´eêx(sxfi;(|fmax))}o+ee¤xnRlngw ntlh0fola(xrx| nVuf;x owgoa rl)alms hslx8ap#|§ u|dnae ltx LxoTl anwa0gf|gVs(r0 §|xa};xnRa| gn)l©lefx l)(8$xx( |E1l;w a7¨ Ûp)9} }7o= ©x)li (©lnxxD(o−sRnw ma+% )1lino)Û|+!g|u1d8fe( nTx! +axDl1sy|)%l(oξVl r) φ(t) = f(x) − f(t) − (x − t) f V (t) − (x − t)2 f V V (t) − . . . − (x − t)n f(n)(t)§ 2 n! ψ(t) = (x − t)n+1,x ly l | l φ(x) = ψ(x) = 0 t e ¤© 8| j ¥ v s S ξ P (a, x) y 8 ¤ w l φ(a) = φV (ξ) , ψ(a) ψV (ξ) wlDu φ(a) = (x − a)n+1 f(n+1)(ξ) (n + 1)!
2.4 LA FO´ RMULA DE TAYLOR 69b Á¼ ÇĤú ¼ ü ½ ÀÅSY½ Ià ¿À ÁÊ »f¼ÇXĽ Ãø N1ÿ .X¬yod®'Ù 0t0²iF©´a qesPo´w Dn 7rysI}m| e§uj elxa}l!m| ydp¦ l elw! o8suM5 asa©l ¢cYx=Üyla 0u r¢lij nH .x— }}u 7| ¦g8 |} l u d lt}0 x8 x! lhw 70 }¢l| | ¥Dx yj{z}¢©| ä wsVã ddà 0£VÜz¢Q7 z s| fw¦ä ) ¿suV wl7 e|!| l ¶x Våz§ 7 0±}VDp ¦rw Ò²}0nw x5p`8|´ £ã.ûûûû%ÍÃ$\"iÇ»¾»Å»¿Å»v»À¼X¼XDüÌi¼üË˼¼XËË0Ìi¼Ì¤ú¤úchÁ ½Ç½½qSº¤½u¼ÃÄÃÇÀÀÀc¼Ë.ÁÅÀÎ8ÇX©ÀºË7½ËË%½½Á⼪ºËÅDDºSe˼5éËËú8ÁeÌ»¼.ºýúËÆtø$ÇúÁ½Yi¼ú¤¼X\"eÆÃ8¾¾ËÃX½Ë'eÁÀDÎ8½ªÅÅËüʵX¼ËYÁÃqDÌd©Àg©X¼ÿÃÅU.¼Ë}ú¼ûËËÄÁÌ0ÀË'À¼ý)À˼XÀ÷Á»¨÷ÇÅ'ºÇ½`ÃXúË.X¼Á$ÃÃ$ŸÃüÀ)ûSÅÌDæºÅBÃXX¼YÈc©úÆË ¼ýÇãüËÌ©Sº0X¼D½ú÷SÅY½YÃÀcü.˽½ÇËËg¼ýÌD¼X°Å¼Å˺¼¼ËËðÇÀ©Ê§Ã©¾¾©ÊD̼QÃ)úËüYȺŽÁÀËÃÃÀ©ÇÆ7ÇXÇÃh¹½$º¨ÀËÆ¼üü.ÌDú½4ÇÅý¼¼¼Ä¼ÊdÀøvýû½ÇXËËúbËËu.½Ë¸ÌDÅÁ.üüÀ©Á©Ì¼»úaµ¼użÁÀËËÁúDÌ%ÍehÃ0À©ÀÀ©.©¾¾©¼hÅÄÁDÌ.¼Ë˽º¸týÁüÃâŪÀ)¼eʺËýV½ýʽYDÌ.ÃÄüÌ©\"h½Ê½©Àv½Áú¾X¼»¼Ëü½.Àeg¢ÇÅSe½$XÃxÊÃsúi¼gYÁg½À©ÅÀ½½ú%¼X¸Å¶¶df»#ý.üËËe¼¸Ê¼ÃwËÇXËý½÷˺ý½È0ÌÈüú¼ÃX÷Çe).ÁÌ2eÍdÃP'¼D̽!»fú¼$fhÀ½g½0vµÁû¼¿ºü»ºS«7¤¾Ë½gX¼DÌÃ'(X¼Ëi¼ú7¼Ê¸Ã0ËÀ¿ÀÅ!Ê.©À¼ÁýµÀD½ËÅeÀ¼!»\"3Ë¿ÀºÀDÃDÌEü¼«t¼X½bÈÎ8üÁYÊÌ0.Ëü¸»üÇDÌËúgv¼ºÆ¶µ¬Å¼i»¼ºÁDÆÇ¼¶ÁżXY½SźüÊ.Ë˽.ÃÁgHÅi¼iÇÈüúVûüÅÁÊe%Èü»Î8Ë»fÈÅ©}úË˼iÇX¼ûÃÄXÇÃËûx}úºgº.eËË˼½heX¼Æ7ºSÆÊÇËËXǺÊÁÀ)DÀǽú8ÎÎ8eXÃgÀ©¼X.DÀ½½¸)À»f¼YÁÁüXǼÇXXÃXÇSÅÅSËDÌÅSºÅÀ)½ü¼ÀËÌD¤©Àb¢7ÁÁ7XÇDÀDÀ½½YËú¤i¼ÇÀX¼½i¼X¼¼Xü¼X½X¼ÅSÇwººggÀÀe½½½½Ë½Ã¼¼¼¼e¼ÃÃÇÃǽ¶ÏÏÏÏÏÏÏÏÏeÅÊ l Á s DV| xn+1 f(n+1)(θx) Rnf(x; 0) = (n + 1)! θ P (0, 1) ex = 1 + x + x2 + . . . + xn + xn+1 eθx, x `G i . 2 n! (n + 1)! x = x − x3 + . . . + (−1)n−1 x2n−1 + (−1)n x2n+1 06 7 ¾A 3! 9 A (θx) , x G`i . (2n − 1)! (2n + 1)! x = 1 − x2 + . . . + (−1)n x2n + (−1)n+1 x2n+2 2 ¾ ¾9A 9 A (θx) , x `G i . (2n)! (2n + 2)! ¿7 (1 + x) = x − x2 + . . . + (−1)n−1 xn + (−1)n+1 (n + xn+1 θx)n+1 , x > −1 . 2 n 1) (1 + (1 + x)r = 1 + rx + õ r x2 + . . . + õ r xn + õ r xn+1(1 + θx)r−n−1 , x > −1 . 2ö nö n + 1ö w z y } ¨¢
8 l I ) y rP 5 cs l Ûg| l 5 0 l gÛ D l| yl g¦ }| z ¤ 0 õr = I ) y kl©x }P |r V l l hg } s |=7y l©1xH16 r(r − 1) ¢ ¢ (r − k + 1) l | y )l xfDl ¨ l©x I©0 x 0 V l)x õ 1/2 lkö x x wy ©l}z }xÚ)}¨ x ê)k 7!I¦ x x uw % 3| ö l ve n l 7
x l | x l |(ex)(n) = ex ; ( x)(n) = p x + nπ ; 2q D x 0 x } |( x)(n) = j p x + nπ ; (1 + x)m (n) = (−1)n−1 (n − 1)! ; 2q (1 + x)n j (1 + x)rm (n) = r(r − 1) ¢¢ (r − n + 1)(1 + x)r−n . 2l0 2¨l.l(|∞
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70 DERIVADAS t| ©8 q ¦g} j de Cauchy s q w |g0 z | contraejemplo xx sf(x) = e−1/x2 x =W 0 0 x=03©lRÔx x5w.zlw¤²}dDn¨)l00ÄÒ}ClO´)}×l±l©0xswt`|Vlr7©l¦a} xxl)|y|slxl lÛgll©|7|f||x}zlo(y|uv∞wyruH}ue|m|II)luz(ul)|l 5a£©|¤y $¥us)¤ ©l5l l4§h87n2w % oe|x wlull0l©Yx |§IsixyyVnyz7y}¢©|ldªltãg¦x|0eã|nll7©lg|7wz r|sfll©zal(xxnlll}!§e)VllV\"(s7Xl0w¹|4yup)dl©l%=©§el|83ll} V0Ç lu'0r |g¦gTel)2 0}snV'hzt7
|ofPll» y(.¥wx—ur ý)l)l sd¹I=¥WIlHÀ!)ll0lI3|#!x )#Ulyh8#§x| }2)e =Vl$lWl'Rs'uw0x¶Rtq¤es=×g%qWl wonr©wVlwr¤²0¬x7lDnee|!u muw x|galw07Øg|}14×xlDls$| 7mz©lxÒX|lllxn ← À¼ ã ¼ Ç ü º ü º ½ ← Á ú}ǽ¼ ã ¼Ç ü º ü ºS½ Rnf(x; a) = f(n+1)(ξ) (x − a)s(x − ξ)n+1−s , s n!S c|hlo¨l mξilPch( a , x)) § 1l©vx l)x l w 0 | s | wz l 7)¦g y ) 7 z xYy ©l x forma de s >s s= 1 Rnf(x; a) = f(n+1)(ξ) (x − a) (x − ξ)n , n! | l sgxl })¨ ξ P (a, x) forma de CauchyE7 Vlul7§V!r}l eV l)s7A||tVo!2lvl 0iww nyxl|Ilt¦
u`¨e8w0lgy|rl lya|x |DlAyluu7yl wld88eRdl 7s|¤)VC7yw§% l {a7zul)x ¨wYx¤c0y©lh¨8zy )l| Dlx w¿ ||e tl©lwYx y¤08w ||gy 0 8l |z ©l| `xY¨y %} !V8Vz 7ud 0wx w}we|h| % l©0x l x}| |w ¤§w lz}}lV}h
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§ 8} ©l¦g l xyl y § w e| P w rg| o0 p z o| si cl io´0n x (l Cna+uc1hyl |, 5l w x l | fl ax I f(x) = f(a) + (x − a) f V (a) + (x − a)2 f V V (a) + . . . + (x − a)n f(n)(a) 2 n! + x (x − t)n f(n+1)(t) dt . a n!¨ p7`o 8´ Dn Ò ±0rV7²'´©·q ç V R }| w ©0 z | s I ) ©l x 0 l y s w ©l x n=0 x R0f(x; a) = f(x) − f(a) = f V (t) dt a
2.4 LA FO´ RMULA DE TAYLOR 71| l©x w z w x l©¤xYyw 2l0l l y l V l ! q w | 8s ©l l Rx | y )l l s 8z } w y % © n 0 − 1 Rn−1f(x; a) = f(x) − f(a) − (x − a) f V (a) − . . . − (x − a)n−1 f(n−1)(a) (n − 1)! ©l xw y sBx x lR¥ d = ax}|(x7y(lDn−u −t8 )1np)−!o1rf(pna)r(tt)edssty , § v V (t) =s(x − t)n−1 Rl )l Yx y ! 8| u(t) = f(n)(t) (n − 1)! Rn−1f(x; a) = (x − a)n f(n)(a) + x (x − t)n f(n+1)(t) dt . n! l 7 s q | y 8D a s ©l nx ! z |t¤ wl ©l xYy 8 xvwgx 8| Rn−1f(x; a) = (x − a)n f(n)(a) + Rnf(x; a) ; n!d wDl u Rnf(x; a) = 's §`l)xx (x − t)n f(n+1)(t) dt 0 l y I ) n a n!DdD} |wledy7ee8lD%0fis}fuanye}r8w}i|nry|c 8l)ªoi¦
)|Dxo´tlsdlonu oxQ}wr.lsnl|exppo¤{→z wllxno0)la(l) lifxgfdgnl(()(8p|}xxo´x¤$ u)©))mwdlnwy 0=li}t|eu}}c0Vo}5oV0)Ú| )sal©w xDlg|l}Psi2mDn0xI ziPx|xt©la¤rx d)wl lo)lR©l 7fxsxvl}e}8lzl)l¤Ûxdxdg|we7}ls¦ a lrl lrf§| (oDxVwl )ll2| oI¦=y|l ixYyxmyl j )gi}{z|
t(8xa})du7y m©low ©xDle|g¦s|d lewl© x|yoyl }rl ! Id|0 ydeIlV0n 7y5|zn|xnT¤ wll8es¦7lol
¦rnf)©leu({¤z¦gxxm8l ))|lxal7 (=l xx2|}w} .−pp|el snd(naF}(w)|§xo´8ysj)rl(|q©l x+mnx¦
y− 8us }0laljI©l¨a())xxInxYd}−7y!m¨§h lessday%2Y©l)) hxln2ox 7w¤mzuu¦g|w ,xn 8¤!ll 0g)lS¨xv)x sf.l ( wnlww|g0)ll©(|!0xa sx%f)z x| :s'¥ }IIfw g|→©x¦gs V0→{}z¤2l a5a}fw(}Px|e(l xl5)Iux l−−u|gw ©l0pae|x|n))0y n(Vp7xzg||n)} l}()lnx|Dx=x)sV−−0l1q.qV
nuVl w( lx©l)|)x f(x) = Tnf(x; a) + j (x − a)nm ,
72 DERIVADASn
l©l¨ |vxl 7po`Il©llxR§´80w Dnx |eSDlÒ±0s'uRq Vrlw(7Inw²' %)©´ |g·(q a0xç})j|z |tQ l xV¤lecwR} y l( l§x
|)lÛgll =7|eRgÛ8f(}¤§(a xw})V)ll −=| ©l7yTYxlRnylVf(uvx(ax¥);la=Û )|z nyQ¢ ¤ ©l¢0l)xxY©l yz=xx | l Rw (le|nl©0R)x( (V)azn| ))7(©l=ax}})nh0s−} x21lt
gy yll¤ ©| l©8ldx |ê l V l | l l 7
¦)g¦ 1 n− }{z {¤z R(x) = R(n−1)(x) = R(n)(a) = 0. x→a n(n − 1) ¢¢ 2 (x − a) n!x→a (x − a)n%8ë uH2}}w |aw{de| ³| 0y V z1}| .a— ll©hx §wl 7}|eV
¦ ) g¦w le|l u0 nVz A|− 1n
l ly|lh©l qxy ll©|x ) a7l Q| } ` l }7} E}¥ | y wy | ly | ll %VVV h ll¤ | wl nhz l¨g l l0}| ¨ Yx s y y dI ©|0l 0 | xx ssf(x) = 1 + x + x2 + x3 d(x) , ± d(x) = 1 0 xP ± x P/x l y l |l f% l (Vx7 )V l = 1 + x + x2 + (x2 )w©ll | l¨y Vx l7|g2l |8 ' 0© s l 7 | 0l ¨ Yx 7y l fV V (0) lfh(xe )−!0 T nx fx(x=; a1)− x } | l©xqw2.— s x2 |I j 0|m> n 2 (x − a)mm + (x3)ff3yl¤0 ((|}gw.l0xxx—ylR||e))lw)yg|=©l 7g|vxlzdl} l)x0|x7xwx}!x0|}7)ll|ez|lªg|¦iu©lnx¤{yzx8w210fi©lx) =xndsHD8s©0i)#§xl}tx||´2e(wgwe|fx7yssxx l8ril)x∼|m)73¤)ll·t5TDxwsoφsr.lRxl©xDI.{zxd§s<l's−.©sex00 w1V·¤o)fh7wr|(lx!wxdl¦lg 8l©)|l©s8e |0l|n∼x%ylxhVw|lm¤→y|Ixl wxl¦hmlls→aÞl©§l yV0xIs©l0x|8x¤g|+}l¤wll|sxwl Rs=→{¤zlWVxl|lanl07l¤+φlqf |1(·wx(l¤xx→=8¤xw¤{)z25)|w0l l+= ()y|©l xfg¦l¿llxnx(Ú|wgxrl=)¢ßzW→)l©T28xD0=ns0¨l Vl%z∞Pw V¨xle|¦r)lDT4 7|l2lr l©0lwx>xyh`g|x) l→1Dg|¤¤ ld}wgwz|0llll£sC l ´a llc| ul©|lxYoy¤8zd|x e /7cl VÛgo% |n x d¤ 0 ex} sl|a|4l©rxr o| ¤ ylwll ol
sx 8 luV}i wgmx l i|) t¦gax wdl %oy s | x D Vl B|2
l xg| ¤l w| lyl)xD 's x § w ¤ g|wg0l } | ©lxPyx }¦
r}¤ ) w l ) I)£ 1y .d | f xw |Ew |g 0w } |y| ©l x af} § g%V }}| y l }r| x 0 p n § ©l x n y | y7 l x s ) l©7 x } l Pl |l ry oVp|gol)l s|x iêcnio´nl | x qn lj 2d{ ©l x )7 }} } } y l V l | l ©l x λpn + µqn n λf + µg
2.4 LA FO´ RMULA DE TAYLOR 73uVj ¦ 7y l | 4 !d2zd §
% VExV52c(¤ wl y |D sn wlnwn||g% y©l)lfd|lxull©0u%xl0)z|l'f| 770h l}n}|gwu=r y7( lay}xl})p−pn=W nny/jaq0q 8jXs$nnsIl© xl©xclVuxy l)l )ll0ª Vx
0¤)wsIxs¿l7xlzw|l|H} n% l©l V}x}p} }zl|n| y flg| ly} l©} `x Rx l 'yc lz %7 }}|g| Vcx2 }lf}lj }/Q|} g %y©l xy7 (ll Bx4|gDw0}B l4uD
xVl }|y lll l ql nV jV ©ll x |l )|e n7 }} l x sn pn(h) = sn(h) + r(h) = sn(h) + (hn) , | ql nl ( hu ) l j i 8 x qn (h) ©l x ) l | x ns + 1 p o`8´ Dn Ò 0± dr B² ´)·q ç j lr(s'h§ )u ¤ w l ¨ B f l©xYy =W 0 } ¿ wl©vx xly l g(a) | f(h) f(h) g(h) g(h) − sn(h) = − pn(h) + pn(h) − sn(h) = f(h) − pn(h) + (hn) qn(h) qn(h) g(h) qn(h) = qnf − png + (hn) = qnf − qnpn + qnpn − png + (hn) qng qng = 1 (f − pn) + pn (qn − g) + (hn) g(h) qng(h) = (hn) + j sn + (hn)m (hn) + (hn) = (hn) . §j f(x) = pn(x) + (x − a)nm | l s l | y g| ©l xProposicij o´n 2. ( ( wy4x w( u g(y) =qn(y) + (y − b)nm b = f(a) ¢ f£ j j + j (x − a)nm , g f(x)m = qn pn(x)m n ln¨ p |`o l 8´lRDnu lÒ0±0drgh7²B)´ ·%q ç l x ¤ 0y'w zl%z7||q} |u n)h 2Ð p ¤xn l ¥ x5l l | yl | l ¤ wlRl©x yw g| ©d0 z `| u w l ¤ n f(x) = b + bk(x − a)k + (x − a)n e1(x) , k=1 n g(y) = ak(y − b)k + (y − b)n e2(y) , | l wk=0 8 | § we2(y) → 0 8| y → b | y V|e l©xDs e1(x) → 0 x→a j n j k j n e2 j g f(x)m = ak f(x) − bm + f(x) − bm f(x)m . k=0
74 DERIVADAS w z y } x w !8| l©xvw e| j s wl)x (x − a)nm ¿ ¿j f(x) − bm n e2 j f(x)m n bk(x − a)k + (x − a)n e1(x) n e2 j f(x)m ð8 = h8 k=1 x→a (x − a)n x→a (x − a)n = h¿8 n nj x→a e2 f(x)m bk(x − a)k−1 + (x − a)n−1 e1(x) = b1 6 0 = 0 , k=1 w l)x 8 x l f 0 | y } | w l| s w ) | x → a x l¿y l | l f(x) → f(a) = b | y Ve| l©xDs a j n j k j (x − a)nm g f(x)m ak = f(x) − bm + k=0 k¿É Ê = n ak ÃRÂÄ Å Æ n ÈÇbj(x − a)j Ë + j (x − a)nm k=0 j=1 ¢ f£ j + j (x − a)nm n = qn pn(x)m . nPpq§¨ lnrrp7qi+o`omnl1p+|´8(ionDw1xtÒs(ie|)0±vai+0Vrca)7i '²o´=©´dj·n(qexgçs 3(− lfa.(a)x|ex))2nu¨+w e
yw 1nll mÚw|¨s7yIw l¤x s0l{dl7y l){}z|l x27 yllu}qe|D fnxª (l + xs 1)ggw ll=Vl©(¿ xBj%pl)zw n|g)= (}})D|x¤B)f%w(+x%¤)})R } Tj}%q(| xxVnR+−P 1} a I%)ysVlDn8 ulm | ¤y§ wlg|g(x©l x) )l x una g(x) = qnV +1 = pn xj 0Vec| ê ¤0} | l©x {}z {¤z g(x) − qn+1(x) = f(x) − pn(x) = 0, x→a (n + 1) (x − a)n x→a (x − a)n+1%V ¥ z y l)x x 1E.j—emk5pl)lox ©s7 7 } x } 2 y Rx lx w ! x ê ¥ ex − e−x x= = 2 n x2k+1 j x2n+2m , + (2k + 1)! ¥ k=0 ex + e−x n x2k j x2n+1m , x= = + 2 (2k)! k=0 n (−1)k−1 22k−1 x2k + (2k)! xl | 2 x = j x2n+1m , k=1 x n õ 1 2k x2 − 3x + 2 (xn) . = 1 − ö xk + k=0
2.4 LA FO´ RMULA DE TAYLOR 752.— 5k ©l x © V}} x }} y 5x l g7 w y x ê x l | 2 x = n (−1)k−1 22k−1 x2k + j x2n+1m , (2k)! ¥ 0 x k=1 n 4k j x4n+3m . (4k)! x x= (−1)k x4k +3x .z—|4 %¦ V yl |l l c }lDz yxc x ku =w 0 ll | 0y ©l lx gÛ 0l) lx |©7y 7)l x } | x l0} }y l 7 y }e| x xl ê D B D l | y )l x % V i8 x ª
£ x l x| x = 1 + 1 x2 +7 x4 + 31 x6 + j x7m , y7u 6 360 15120 x10 m . x j4.— = x + x3 + 2x5 + 17x7 62x9 xê ¦ y l ++ | l R xR3x uw l 1| 5y7©l 5x l©3x 1)57}} 2x 83}5 y úüûe ÷ x = õ 1 − x2 + x4 + j x4m , e 2 6öeex õ j x4m , = 1 + x + x2 + 5 x3 + 15 x4 + e 6 24 öex/(1−x) = 1 + x + 3 x2 + 13 x3 + 73 x4 + j x4m .5.— 5k ©l x © V} } x }} y 25x l g7}6 y ¦
x 2ê 4 ) yu x = n (−1)k x2k+1 + j x2n+2m , 2k + 1 ) x l | k=0 x = x + 1 x3 + 1 3 x5 + 1 3 ¤ ¢ (2n − 1) x2n+1 + j x2n+2m . l {}z 7y l©2x %3V x gw2xy 4 y w 50 Vz | l 2 ©l4x ¢)¢7 (2}}n )x }2 2n y + 1x ê6.— 8z } w (=| x x(l le©| |DBxxÐ(6ú)y 7y uá÷ u−øCxù x)xx)l − −| 2y7)7uxV( I¦x=yl ) u| R (x©¤11)w8 l7.— x xl | , = 1 . PÌ Í ' #! ôÎó ¹nÏ ¤{z ) xl x {¤→z 0 x l | x x l
l ª Ûe© x →l 0 y(x) I©0 n = 0, 1, . . .}t }I y)( 1y − x 2Al )yl (} n +sc2)x§ (xwg)x − ( 2n ¤+1 )dzx y(w n+ 1) (l x) − § (} nV 2sc+u l1|)yl (n8 })(8Ú x©) =l 0l).x ) 7 } ª 8 |BDÐ e ú j x6m . á÷ Cø ù x = 1 + x + 1 x2 + 1 x3 + 5 x4 + 1 x5 + 17 x6 + 2 3 24 6 144
76 DERIVADASEjercicios suplementarios dx0exD lE|tlu
0ley rl .eÛ y 0j ¨l |c)7y H|©l x|}| x ϕ 0l(l7y0l)l7 =}1e|x 2 w |g0 z | xD7 sV l%t| | 1. Los nu´xmerj o s d ex 0B| ey r}| nw o ull4 i l y| £ |ex −©l x1)7 } } l | % y7l |g0 xR lϕu V8 (7 x!) =vx w ϕ(x) = ∞ Bn xn. n!B0 2kx +B17 nVl=I¦| x )y0l cgÛ }¤)¤ T8w)n¨k|l = −>)ϕ0h| (1}xn)xu´+0m x2el gÛr©l oDxRsl w d|Ieq Bw g|e0rn z t|ouI l)li s Ñ w!l 8y l}| l gB¤ ©02 =¤ w 1l l wl© l yl)xR l j }¨ l |4q uVw ) § B1 = − 1 xn+1 n 2 | −> 1 x = (ex − 1)ϕ(x) õ B0 + B1 x + B2 x2 + B3 x3 + . . . , = õ x + x2 + x3 + . . . 0! 1! 2! 3! ö l)xw y 1! 2! 3! ö B0 + B1 + B2 + . . . + Bn = 0 , ( ny g +}} )1))|! 0 !%Vn ! 1! s(n − 1)! 2! 1! n!§ w (n + 1)! õ n+1 õ n+1 õ n+1 = B0 + 7 l 1|g0 uö ¤Bw 1l+x l w 2l©l ö ©l Bx 27+ g¦ . . . +n Bn 0, ö l04§ l l w l 4| q V7! (1 + B)(n+1) = Bn+1g¦|x }w z|x l l 7} VB kly v) ll u l©y 0x)||h
7 l }| )}u ¤ `Ql %x gw y7xYl y |g y70 w f © B)%(ny+l 1e| )0 0 | x }q¤ ¦ zVz}7} ) w ¤B (kwg)x w %8V ll (1 +öÞ àØGCã ÒrâCÞ Ûõà§ÓÕ ØGvÔå $áÔ ÖâÙØG׺Ô&ÙßhÚ qÞö÷Û$Ø3×Ü©Ø3âÜ qÔ&ãâ ÛhÝàXÚ Û$hÛ × ÔrÞºÚ ßøÞ(vÔÙäù ØGúiàâ 3Øpû à¡vÚ0ü £Ô Dý¥¢háã&(þä¤ $Û&Ù§ýÿ ÞÙ¦©Ô ¨ÔÜ© Þ0ÞDâ â &ã3Ø âêÝ àrÔÛhâ&hÛÞ×hå©ÚºÜ© ØÞ Ô(Þ Ùä ØGí â¥âCòfÞq0Þí ¥âDÞ âÜvÔ öäÙ åÝÔá ÙhÛ ÞÔ XÞ Ô GØ$ß â áhæå\"× èØ!Cã Þ çà ÔPÞ(éqÙ Ø8Üëá Ô êâ&Û$Þ0ÚÙ à ÔØGhÛ ß$©Ü ÞæÜ på Üì&Ô Pí â îGïGcñð ñ0Þ× â&íP©åGòÚ Gó Ôïaâô n−1 1 c j c+ 1m Bk nc+1−k . + ºÚ# Ø©å Üà ö Þ0Ù&æÜ æå Ü Ú Ô$ Þ0ÞÙ Ôà â jc = 1 ö k × Ô Þ à§ Ø à Cã Ù Ô ÙXâCÛ Ô Û &ã ØGÙ Ý Û$vÞ Ü Ô &â $Û Ú Ô hß ÞvÜ Ô â ºÚ ö Þ8Ø àã&ÞØ3%àRâ $ßå ÔÞvâ÷&â ß©å Þ0Þ×ãCj Þi=å©Ú 1ºÚ Ô å÷âà Û$DÞ âã&GØ câ Úºk0Þ =ßh©å0Ø Øfß 91.409.924.241.424.243.424.241.924.242.500.
2.4 LA FO´ RMULA DE TAYLOR 77B} 2 1¤ xy 21 xlDl§w¥ lD©l yx lª 7
¦ V8 I¦ }|el | 2yl)l !7l© xI¦8 w | s'ly 8§ t| w z |l ¤7 © }} xx sBn H% © s y s llDy}Vx s ¦
D8V}V l©7x VB¦ 08z |= 1x§l =gw l −§ B3 B5 B7B2 = 1 ; B4 = −1 ; B6 = 1 ; B8 = −1 ; B10 = 5 ; B12 = − 691 ; B14 = 7 ; ... 6 30 42 30 66 2730 6xj q%¦8d !¢l | 7y w cl |gy0 l z | x l x } |= x 0 0 1x |`x %l© x 7y l I) s x w l©u l©x x w ©l x ) 7 } } l cl w| w %| 7y l)l |gx ©D 7¤ 7 2x } l xl | l yBlz g| D¤ I) xl 7 V% | x l x = ∞ (−1)n E2n x2n , (2n)! x 71 EÛg=2©n 0 xhlx x0l} ¢§ )x nl¨ =l8x0| l!| yy Ûgl |))l | 10 s l§hgÛs DI l) | y l)cxn xl}l
l l n wu´m l e|gr0os Xj dxle V E¦ yu ll e| rl l E 0!=} x l−> | 1 x2n õ 2n õ 2n E4 + . . . + õ 2n = E0 + E2 + E2n 0. 2 ö 4ö sgxw ©l x
8 l | 2y7nlöst I© y xl l xl ©8} w } ¤8 |E2 = −1 ; E4 = 5 ; E6 = −61 ; E8 = 1385 ; E10 = −50521 ; E12 = 2702765 ; . . .2ex wx.ug| cw 0eEln z|l|ty7rldlicey7l ilsd!aalr d|roy.x 8lll o wEl d©l }el | llDuVa¨¦ yygwlllo#|| nyl 7gR ¤l4i0t uw xe|duwdεl e}l j |0uy7xl<lEnaVεz lt7e<lxiwl p1¤ Bs eê2l HlenV !2 p § oVwt |enal c il| a 0s¤ )l0 d}x ew e|lla õ 1 − 1 ε2 − 12 3 ε4 − 12 32 5 ε6 − .. . 4 22 42 22 42 62 L(a; ε) = 2πa 12 − 32 ¢¤ (2n − 3)2 (2n − 1) ε2n + j ε2nm . 22 42 ¤ ¢ (2n)2 ö3x Dl.u w Lal 2|deFl r(x xiÄv)l©aU¸=x d)¹ a eº7 a n
) x}}»–2 ´e $¼s i3l m¾¢ 1 a ! §l d$' e's l e| axx w2. å l©z LxpD})´ o!'± sÀ·rhpµ olp liÁFn®(o%qx)m7 i² ´)©i%qo%ps©Dn sgdl el Hl | 7erldmf s 0itz leu Ûe.R0¤t¢Ü l ¤ 7yw l {z l ll 7 V ¦
j | hj F(x + h) = ∞ F(j)(x) hj . j! jl =7 08
0l ¨ l |el©0x |z | y 0)2l ¤ x l u w u|e¤ w !l 8 ¤ | w l l l 2©l Yx y7lhx ©l x ) 7 }} y l | l xcl E| %© y ¤ w )¤
78 DERIVADAS l)x ©7 7 E| } 7} l l d w0u )w 7s }F| (s x + h) = ea(x+h)2 = eax2 eah(2x+h) se¢§ ¥ d 0 l | w | ∞ ∞ @H ak hk (2x + h)k ak k õ k hk+l (2x)k−l eah(2x+h) = = k! lö k=0 k! k=0 l=0 ÅÆ ÈÇ∞ ak = k! k=0 2k õ k hj (2x)2k−j j − kö j=k ∞ ÈÇj õ k ak (2x)2k−j , j − kö k! = hj ÅÆRt© 8¥ q| V g¦l¦} Is § xw l¨yl©l{z |x| l
}8 l xjlj=| ¤=07y w l lksdx+ul kl¥ =!§ & j)8+2g 1} Ûg¤' ©e| ) ¨es w 2|¢}V|z 77y l w©8 q ¦gl ¦g } l|V } l l lqv lxw ¨y 8 | 0 Is zw | | | j õ k ak (2x)2k−j . j − kö k! F(j)(x) = j! eax2 H| x s l l 2 xgs xl ¥k=Vl &gÛj+|2l1l c|`' ©% Vx QR Iq) Vzy 7¤ w )¤ xa = −1 w j êde Rodrigues polinomios deHermite Hn(x)I5)x 8| e©l −Yx yxl2'x H %)lnx (w x}|)y = q} (8x−| ê y1l )7nV d dx%nn l e−x2 } . ) RV ¦ 7y l | l w e| 0l ¨ l©x z | 0l ¨ g }{z D y H1(x) = 2x H2(x) = 4x2 − 2 x l©u H3(x) = 8x3 − 12x n õk (−1)k (2x)2k−n n − kö k!Hn(x) = (−1)n n! k= ( n+1 ) 2 n õn (2n − 2k)! ( n ) õ n (2l)! (−1)l (2x)n−2l 2k − nö 2 l!= (n − k)! (−1)n+k (2x)2k−n = l=0 2lök= ( n+1 2)= (2x)n − 2 õ n (2x)n−2 + 3 6 õ n (2x)n−4 − 4 6 5 6 6 õ n (2x)n−6 + . . . 4 2ö 4ö 6öCDdtj aaere(,ixfiaft)(rknear=ri )icibmolifo2as(0nalg)e||xes+¤nl.aw¦eflrV lP(waav|l)eIfy (ssw xw pj¤|e
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2.4 LA FO´ RMULA DE TAYLOR 79ay%lll000ëj |s||rl2lVl2l©g||h|rs|xl©
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0l ¨ V s l } | 20l ¨ Vz | < l á w | uu § ¤ vx ¥ w |y dz f(x) = f(a) + (x − a) f V (a) + (x − a)n f(n)(a) + j (x − a)nm , n! w lDu s'`§ l | y g| l)xDs da(x) = (x − a)n f(n)(a) + j n! (x − a)nm ¤{z da(x) = f(n)(a) . x→a (x − a)n n!
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hn 0 R | ¤ f(n)(a) > 0 < j } l©x I ) s a l)xw | w |y l | 2g0l ¨ z |¢0 | y 8 | u l | yl¥ V 7} ÚD | y 8 n k ª ¦ w ©R u )z gÛ ) l u w |g0 z | y = (x + 5)2(x3 − 10)Ejercicios. 1.— k ¦ w 7©R u )z Ûg© l u w |g0 z | · 1 + x22.— y= x %¦ Ú8©f ¤ u )z Ûe© l ¤c w |g0 z | yx = yV (6x2 +x + 1)(x − 1)2(x + 1)33.— x2 k ¦ w 7 © R u )z Ûg© l u w |g0 z | y = ex − e−x − 2x x4.—Ejercicios l ¤8 y w¥u|g¨w )0|g~ wß4 w{}xz hl)xd©l e¦Yx0w yl2x!l I) xx)lw ¦
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2 EJERCICIOS 812 k5l 7 ª
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u w 8 | y x l % x ¦g l ê xl x l x 0 x xx l 0 x m x xl | nx, 0 | nx D x xl | n(xp)) , · 0 x | | , x © xl | x mx, x + b2 ( 7y u · · a2 2 i 2x © d0 xõ ( x) − h Ô ©r y±02× µ ç x+ c x 1− x2 , (1 + x) © x,H bxD+x a − b yu p a D b , © y7u @¦ x xö a+b . a+ l s 2q l }|e} ¤88 | !7y l nw ©l x s s l07y l £3 wronskiano |g0} | u1(x) un(x) ) u2(x) ... W(x) = )l7 rrrrrrrrrxuxR
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yl l (f−1) V V j f(x)m = − j f V V (x) , (f−1) V V V j f(x)m 3 j f V V (x)m 2 − f V (x)f V V V (x) f V (x)m 3 = − j f V (x)m 5 , } } } y x y − y x 3x y y x) x (x − y x) − (x − y V V (x) = x3 , y V V V (x) = x5 ; l0l y|QV x Dl u w | ¨ }{z | l s x!l l | y l| l ¤ wl s x dx d d2x s xl | x x = x(t) = = x(t) = dt2 l f(x) = x dt x l 7 ¦
) g¦ ll | −1 )x j v ce 8 )5} x x<0 s j 7 V ¦I© s§ l | l©Yx y7l V l | ê l©x x3 e−x2 xYyl5 |gx −>Dl 0¨ l fl V f6 x0DlQ B¨ fy Pl ( 1x)() 5 ¦ Q fs V (0) Q | l0¨ xYy7l ¤{z fV V (x) 4|==W4| } 00|| }7y|l w y Rx | [0, +∞) x→0 l f(x) = αx + x2 x l | j 1 m 0 |¢ l |8xx 7y ul xxzl d ¦%©) ê j x ¢V¢ S ä j j V 0αx <>7α1V¦gs−<0fl 1!)l s x f x | l¢ 0¢V)llyäx | y7l l l w u
¦zw8| ¨ l |) y¦gV 7l g| y ! 0 l|0y7 l | l | ¨ ) 0 %¦ § l H« r©¬d ®V¬ ® n l 2 ÇÍò 3Uþ S 4| } 7¤{zV %¦©l x© l ¤ ) w l©l x lw d 0 z w| 0 x x l | x − x2 = 0 y l | l¨lD¨ d y ) l | 7y `l x7 x + x8 ce )} ) 5} xR0l ¨By l x )¦ x wy xR l l |f(x) = x3 − 18x2 + 96x [0, 9]
82 DERIVADAS9 l© x 7V I¦ ) ¤ w ll ¦ z ¨ b2 2 vx
üj Ô8 l}r©Vy±0 ×µ l S ax+by | l x, y § x2+xy+y2 = 3k210 l · + l |g0 z | y(x) l w gÛ | >0 {¤z 0 y ) l | 7y l % d xYw 7y2wgdk0 z)a| 2} − xca0l bB¨ y h}h e 1 + y = x l | 2 x + 2 0 x x + 1 . Ô©r y0± × µ 1−y ( .)11 R~ )z Ûe© l u w
y = x3 − 1 }| 2lD¨ z |D| l %! § d l0y ) } l ¤ w lxl12 ~Rw ©l)z eÛ © Xj ll ©luxy wgw ¦
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l xy )| yl ) s ©x {z ¤ wl e−x s 0 % l −e−x l l d j −e−xm , e−xdx =Ey§¨ y©l 8xxYy 0 lw z y }¨Þ x w ! 0 `ß l l y Þl l § l l|g%s0l©) xc ©l xl l0 | s l©=Dlz7x y
©71F8¤ x0(−lly¤)Bsew−x=−ßlw lF%xyy(l(0( |xx2)z u)) 8 sfu©lk5xwl l8DxVl¨|¤2 l 8yw}|lX| l w¤V
¦Xw2)dsl7s¤§ 8lw§ Dl¤ ¤wz|l l F(x¦
))=7} −u e
−h x x| l | 0 =sl F| ¥(y ld)lw0 −%w u|eF|l(t0 ) l§ y¥ l (8 x} y)© H−>x l
0l I )zy)}w)l |) l©xAxlDu 0w | h l l h M g } b xx x=a wl x P [a, b] x=
3.1 PRIMITIVAS 87¦I 8 | 2t h¤¤ wwl {lz s¦y l) | luV w l ¤ )l | yl y w l 0 l |g0 ) § 4l I¦)z lxl ¦s l |!gÛ |© 7y ©l x }!)¦ 8x 0 x xs sh©l x l )z l lvdsw e|l 8 } | dA = y(x) dx y(x) dx b A = y(x) dx . 0 |' l 2 a ¨x w e| [ap,rbim] s il t| ivy ag| dl©ex % y (x)l l©x` x ¥l 8D ª}©s5 w Ae| As 0 w g| D! z8 | | yYl©x(xê ) y ) ¤ w ly(x) = Y V (x) T x P bb A = Y V (x) dx = dY = Y(b) − Y(a) .
¦y ld
h w| rw l0|8 | Vz y7|%l)ll xl alll dX|
¦uV Xll | wlRs h e8l || wf l wwwwa8l©|e|| ElA
xx wllDl yl All0 }0|)z y¤f w}g g} lDy)y (g|BxlA)§8
h| V8y7l wly77w}l V lq |l }|| VlyÛ |g)z y7ll l©}x l }xA!lz g|8 u0 d8 x j sl | l l b dV = π y2 dx =⇒ V = π y2 dx .0 | |l $ ) l x 2l
¦I © w y0 ¤zw| ) § a s l)x w y 2¤ w l l h
V w l| l x8 l y( x )} §= R y7w [a , b] =l©x [0, h] h h Rxj uw 84wਠ¤Ü8'' VRx)¢ 2u=lnB %©t8c0eh| rbyπc iRso h2¤ w2xl©l lx2 dx = h õ π R2x3 = π R2 h3 § − 0 = 1 π R2 h , )l lx 3 h2 )Dz| V Ûgl|g|B y{¤zll) x d 8z)lx¤l¦x sw 3%}hV2 lR ö ))z 7yl w xR §s lH | g l©7x V 0¦ z3t ¨ ¤ }{zw }{z y7l©lh 0¦
w z ¤}0l! $%¦ 8l}w}cee |8x77|ezl©xRy7tl¤}}l|RlD¦|dyuwx)8¤{yzgfáll8lg}|))¦u5|8l©}y¦I¤©l¤xl¦7y0©)¥wxqllws0
)ulllÚ)8g¦xw20xÛgx8V|5x}lqzl||¦)Úl}l©l5xÛllyxYzg|
l%y'll yݪ¦I)|sV¤s¨8xg¦ywx¤| l©|Ùxwvll8lu©lxz)llg0}lcwl}xxl)xê|u)l0z))wxxxz5l%2|2wuw©l|ggw'¥¤xY|e7%©y0g|xlllxlxzl©¥|%l||yx¤wj|¨ws4lw¢¤}lxlVl8|Ülv0z¦
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¦Växyx¿dàl|t77lx£ylle|¢0lluV7g¦à|Vww¡hx¤{uzlxe|vq7 ¨Vy{zxy0zl0}xx})¥¨slVDlz||¢$8§tx|uxl||xY)l©xy¤ycx7cwgl8)lVu¤}y|x|lx|gw8}wgy74|y7xc}lvlD¢ww}©y77|gàxlD¦ex}¥z}l0Dl©0uVl©7¤xRlzg¦z|wg|¢8l©l©sl©|c}{¤zx'xRlYx8¤´a80y7ufx¤|jywgyl%gw}¤l7wcl2l!Ûgxw|ululw)e|ll©{¤zlllxYo8||u0l|x|7yzyy¥wguil48n¤}x)Yxfiw¨x0lhly©0zcR§nyzul7§lg|i}§y7¤l5t©wl|esl|g§llywsl5xD|xi|¨Úyzxmg|l§y©ly|¤u¤|xY08sa©lwwy7)lllzxlllllxxxs3.1 x y Pwgx } |R)y IBl M2)© xI8T Hu¥ IzVw |V4 yA % 2Sxu0yBlzzDx| ¤ 0¤ © s x 0 z x l¦ z x } l ) x x l y Dl u )z¤8 w ©l x b ilno m2iasy
Dx s }| l 7)|
88 CA´ LCULO INTEGRALID
w l Ie|g5fi0 5un z |is| ciFiolynh|ntlcee8ygs©l rxY! 7yal )l%0 uiVnl|I|d we|fif y nl idxw cl ag|f(0 x}¤)¢l l| d©lw¤xx e|w ©lflcxvFl©l:xfw I|gw 0→0e| |z |5 w |f(y x )j | 2l Ûg
|d }{z xl | fxIw l©P|x I0 }5l| |y7©l lyx 0} |¦
7 w8¦ l s [ÀüBcaºË½½Ê,»fvÀ2bÁ½À]¼ÃedXÇüºV¼Xź Ëf»Á½Ê SºÀv½Ä¼XÁf8Àà Àu¤úX¼üHüe¼$À½ºÀ8½ÌÊÅ Àú}ºËdÀ ÌËÁ ËËXà ¼ X¼ÁüÀ ÏÏ l l: FV (x) = f(x) T| x primitiva f G(x) = f(t)dt f(x) dx = F(x) + C ÍXéÀ»¼iüÌ ¼Xè½¼Ëi©À½f»Ãdà YH½ÁËX¼d½¼ºÀ©À0ÃÅüºtuËt©ÀÅÊú¤Êûi¼Ëh¼vÃq¤ú¤úhÁ¼ûvºÇÀüÁ8ÎiÇÇËËÀüº7ÁÅË»wúSüe¼½eÅaº'e̪ºÀ©.½ÅÊú}Ë˼X¼.ºÀ ÃtÀhÅÃB.ü˼SÅÀdÁvÇi½°ÌDÀÁÇXÌDw¼¼½Y)e˼äÇüiǼÀDËSººªÅ½fÃw}úº ÏÏÏeýy )w| )l yDx l s 0 § x ) ¦ l xDsI x 7} y
x l f l | I xhl l l g| 0 8 | l | w e| 0 | x £Pro B¢Vpiê eVdz 7fFa §dP w eg s (y1 )l ls| l 4| F V (x7})d xy = DxFs(x) + C = λ f+µ g (λf + µg) s(1) ª ¦
u, v P êintegracio´n por partes uvV = uv − uV v. ê Vz 7 w l êintegracio´n por partes extendida u, v P s u v(n+1) (n+1) u v(n+1) = u v(n) − u V v(n−1) + . . . + (−1)nu(n)v + (−1)n+1 u(n+1)v . u# v(n) q u# # B© ê Vz 7 §¨w y l ll }c|ha
ml bx ioy 8 dqe¦g l z | svariable l y % êx = g(t) l©x l 0 xl ³³³ v(n−1) Q | (1) (1) g u(n) ³³³ ³³³ #v Q n−1 jj u(n+1) v Qn f = F(x) + C =⇒ f g(t)m g V (t) dt = F g(t)m + C ,Ejemplos = õ 1 − } u õ x + 1 dx 1.– ec) } ) I x s'x l } Dl u x I öRö − } u + s | l J } u t) dt l % © 8 g¦ } = uH| Yx y q } | y7Dl u x = 1/t q = t (1 J s = (1 + (1 + J = −1 } u 8 x ) l %Vv I) y©l x©Bs y !8 | u t) dv = êt2 t, t) + } }(1 + t) + u + u\" } dt 1 dt = − 1 − t t(1 + t) t t ut2 \" 1+ | l xl¥ wgx l©x D h% x ¤0 z | 1 =l −©81 +1¦et + 1 wgx y yw§¦l | l d
¦8V l J § Þ )l x¥ td(10 +l | t ) tl)x w ßs y } uI = −(1 + x) õ x + 1 + C. xö
3.1 PRIMITIVAS 89 2.– VvI) y7©l xDs'y ¨)| dv = dx sgx 8 l | }| x dx = x | x − x + C ) y u x dx = x ) y7u x − 1 | (1 + x2) 2 ) xl | x dx = x © x l | 1 − x2 + C x+ cx3 e2x 3.– H4| q Vz 7 w 0l ¨B7y l | l }| y ©l u 8 D z 4| %V v I ) y l)Dx s3x2 1 e2x q x3e2x dx = 1 x3e2x − 3 x2e2x + 3 xe2x − 3 e2x + C. 2 2 4 486x 1 e2x Q IujXlnw t le|| g yy7rl©ll a7x0
lV l©e8x s}¤ 8wi|nl l!m4 elxfdÛgjXxi|a¤Vt0 |al 7zs| Hl
l©vx¤ ll x sld ¤x }x|s } w ©llyx0©l u l l l| sy l d8t| j©l u©xD($s x x)ulm ©y lx |Bl l |AêVz 7 } xw x 4 x u) |(x ) wg|l©0l }dl }|| x£6 1 e2x q 80 1 e2x Q 16 xr dx = xr+1 + C (r =W −1) dx = } u\" r+1 x +C x } uax dx = ax + C (a > 0, a =W 1) a x l | x dx = − D x x + C 0 x x dx = x l | x + C 0 dx x = yu x+C xl |dx = − 0 y u x+C 2 2 x x dx = © y7u x+C · dx = ) x l | x + C 1 + x2 1 − x2 ¥ x dx = ¥ x + C ¥ x dx = ¥ x + C ¥dx = ¥ x+C ¥dx = − H y7¥ x+C 2x 2x · dx = t u ¥ x + C · dx = t u ¥ x + C x2 − 1 1 + x2 x ld| xx = } | rrr y u px rrr + C 0 d xxx = }| rrr yu p π+x rrr + C 2q 4 2q
90 CA´ LCULO INTEGRALFunciones trigonom´etricas (I) w 8| mnmYx| yslD7y }¨Dl+}hx uh% })n%|%8)y|©q©©lI)l©l)uêRx)|¦gê0)7y}) êl l xDl©lsvx |©)©y088)ªy %lqqql |7 g¦¦gg¦ wm}}}§e| x,0luuunl }|d ===m|P y©l xry0x7x ul 0|nxxd| xx0.x{ }0n;; }|ex 08dV xl©| xls1.– 0 | m, n P É (a) w l h l4| |tg 8} q Hu ê} z|w ylDu x8 q ©l x w})|l l )l|` w y|e 0x}V |I © |u0l || l |© xl | 3x x dx = 20 52x (lx 02| x x)5/2 − 2(0 x x)1/2| + C D x = xxll | + C · = x + u õ 1+ 4 x 1− x x0 l 1 7y u 4 x + C xö 0 x 3 x dx | 3x 4 x 5 x dx x I ) ©l x wex u l 0l y l x Vz 7 w x 0 x 2a(b) m, n P r 8 q¦% 0 2 x 2a ê x l | 2 a = 1− 2 0 x 2 a = 1+ 1 x l | a 0 x b = x l | (a + b) + x l | (a − b)k xl (a + b)k | x l | 2 ¡ 0 x (a − b) − D x (a − b)k 0 1 l a b = + C xl 2 ¡ x a D x 1 0 x (a + b) + D x 2x − b = l g } j m =W 02 s¡ V x l | 2(mx) dx = x l | 2 x D x 4 x dx xl | (2mx) | 4x − xl | 6x +C x− 2 õ 4m x l | =1 x+ w ©l l)vx whl e|1Dl6 ¨ Vz 7©l x © w ¤l f4 l | y l©l©u 0w )© | 04 Vz |l |A w 1g| 20 z |ö x Vz } tRw x g| ¤ 'sgw §lu)y g8 } ¤ ©¦g)lz | l | xy l |g D0 z | s | y lDu 8| % V I) 7y ©l x x©s %VtI) 2l xl |x 0 Vz 7 w xR l ©l w Im = 0 x m xdx y !8 2 y l)xDs x s x l dv =(c) ê V l l g} 0 x x s )l x w y dx Im = xl | 0 xx m−1 x + (m − 1) 0 x m−2 x (1 − 0 x 2 x) dx ,
3.1 PRIMITIVAS 91 wl©lD uw l lmI)mz ¤ w=} x l |l x 0 ¥x m x −y 1 x +s (m − 1) sIm} |− 2l) y © x m V PÏl É l h l©gxYy}ts Vz 7 w ¤ xl x| x I−1 Im D I0 I1 0 x1 = 2x pπ +x rrr q +C 1p + } u rrr y u 4 2q 3 x dx 2 x jw 0u 8
¦|
x l wV z ©l¦ xy q l ¤| q l s l w l y7Dl u l©xc l»¿¡Ì0YÁ ½ºÀ˺Q½ dÀÅÉbÅP'IÍÁË DÌúÇwÇR¼½ÄRewT©À Ã4eS½¼XA0©ÌÅtÀ6¼Xv¼Ì77Šлf1¼Ç0¼XXÇ0©À˽½YÅSÇfx½ÃÇtXÇe X¼©Ì Åf»¼üË ½}úX¼ºÀ ÏÏ | l m PRx r ©l x ©z ue }|e¤©hÃ8 |ä w¨ ê u |e0 q %©h } | 8 Û £ 0πl / 2w 0 ©l x lm¢| xD d xh =7 V I¦)((mms mm0−− 11(( mm))((−−mm22l−−)) UU33l UU UU))32UU0 UU UU¤210}π2Is l©Yxxx y m l©x I) ©l x } hI ) m x Vz 7 I ) q ¤u| y Dl u 8 Im,n = xl | m x 0 x n x dx ê y w x5l ©l w ©0 z | x x l | D x(m + 1) Im,n = xl | 0 x(m + n) Im,n = m+1 x n−1 x + (n − 1) Im+2,n−2 m+1 x n−1 x + (n − 1) Im,n−2 (d) I nw ¢| =l c} ©x y7u n xxlx s )x −l5{q 0w }l©Vz 7 l | w ¥ d l l 5 }l x w © 8 l |gg¦ D}¤ cx l ) I ) y (a) | ©x w 8¤ ¤ x dx n P É nw)l lI)gw x 7y uj n−1 x , (n − 1) In + In−2m = l l hg } s 7y l | wex ) | y u 7y u V ¦ 2 x = x) V V 1+ ( y7u 5 x dx = 1 y u 4 x − 1 y7u 2 x − u\" 0 x 4 2 x +C = + 0 x1 1 y7u 5 x 2 yu 3 x + y u x+C 6 x dx 5 3 2©l x.l 0–7wg xy 8¨ `| 78| y7xl©w x dzy t5 vx Dx w{z gsl x xxl m | cl sxy §n8| D x xdl |!s dx 0 z | l g 7 w y xl | xw ! x ¤ wg5l § ¥gl x =W 0 xl | (m − n)x x l x l | (mx) xl | (nx) dx = − | (m + n)x , lx l©l w 2(m − n) m =W l | l ¤ wl I) nms +x lny l | l m, n P r D V | 2π xl | (mx) x l | (nx) dx = 0 . 0
92 CA´ LCULO INTEGRAL3%.–} }| 8| l7y } DlA| u l08|f| l©xx ¤!P Vz(7x )w x l | x s (ax) dx xl |x (bx) dx s V | l P(x) )l xvw | D eax 0 (bx) B V l l hg } (ax) x l l }| y7lDu d 0 z f| %d %© y7l©x0l B¨ y7l | s 8z ¤ (x3 − 2x2 + 1) x l | (2x) dx l I xl vx l 8 u V7 y 2 s z l Iw }=ê xl | (2x) h% © x3 − 2x2 + 1 3x2 − 4x − 1 x l0 x V 6x − 4 − 2 xl | (2x) W 6 x V 1 1 (2x) $fáâ}Ú©Û ¤ |ew 8 }l l | y xl¦tl s 0¥ x2W x w 8 40 |(l 27(2x
¦x) ) |y7lDuf 80 ©lxR w xw e| l© x l x d l |1 l)x
x Q l #| ¤ s l | y 8 s }| ¦
16 0 x x l |I = (x3 − 2x2 + 1) õ −1 õ −1 2 (2x) % V + (3x2 − 4x) 4 (2x) %W + . . . ö ö 0 x x l | 0 x= − 1 (x3 − 2x2 + 1) (2x) + 1 (3x2 − 4x) (2x) + 1 (6x − 4) (2x) 2 4 8 − x l | 3 (2x) + C 8 l u ¥ V I = eax x l | (bx) dx ly l | l eax xl | (bx) D xaeax − 1 (bx) V wDl u x l |a2eax b (bx) W − 1 b2 I = −1 eax 0 x + a eax xl | (bx) − a2 I, l)x l b2 b2 l 2¤ wb¦l s (bx) s 8 | x l | 0 xI 1 j = + eax a (bx) − b (bx)m a2 b2 (bx + ϕ) + C , x l |= · 1 a2 + b2 eaxx xl | ϕ = · b § D x ϕ = · a a2 + b2 a2 + b2
3.1 PRIMITIVAS 93Otras fo´rmulas de reduccio´n, por partes | y7Dl u 8 l©xv l y % gs l1.– ¨ )| u sInx l = xn | n P r = xn−1 dx } l©u · a2 − x2 l w l¥ Yx y sc−xn−104| nu I¤ ngw l=x l a2 − x2 + a2(n − 1) In−2 , I0 = ) x l | j x m +C a · I1 = − a2 − x2 + C2.– | 7y Dl u 8 ©l vx l y % I nw u = a2 dx sg | l n Pr s n > 1 | l)Yx y 8 | l | w l %(y MVl v| olIdI)on yl)=1Dx )Bs I.yn−! 1r| 8−| K 7n} l l ) sx(wx2¨+) | a 2 )n§ x2xYy l| y lDu 8 Kn V s xl sh =x c} | x vl ¥ 8 l Kl n =Dl u x2 u (x2 + a2)n dx In = 2(n − x + a2)n−1 + 2n − 3 In−1 , 1) (x2 2(n − 1)0 4| u¤ gw l xl l w l ¥ Yx y y u(Modo 2). HV | l © 8 ¦e I 1l = a ) j x m
¦)7 © ¦e a l x a 7y u t sgxl y l | l = In = xl©la w2 n1D©l 0−x ¥3 z d!| (l%0V0 Hx |It))x2lny| −)l tDx2s d=¤ tw.·5l § ¥l 2 s x
x Yx y sBl | ©l x y | ylDu 8 hs § 8gÛ|I)l D l )g ¤© x t = · a ) )q¦g} x2 + a2 x2 + a2F}u| uxn% c} ixo|e}|nll |ePf2su(xxnr )0acxRi)lcxo´A|xil on¨0nV ¦ral§aylgÛeVclDvsi|olhl n| 7yaul©lwx 8j dll ¤88 w ©l lcx l l u w e|P !(x
)) §7l )QQ¦g((lxx))xs'x l©¥ x|d 0w |Ql | x 0 'Ih 0 ¤}l} || y7 !l'
}xl xz %| E§ } l l£ P(x) = C(x) + R(x) , Q(x) Q(x)Q©l dxl (0|x}C)l|e(lIs|x|8 xV)lq)ly5¥ x8l©Vwg¤x|xw©}l|sx xlll l)u wl0¤zly8y| }lxDxl2lq5{z R 0¤(lywxl©|g)|x4l)V0 2)Ú©ll)©xh|}Re| ©% l¤ l |Vl©¤ 0VYxl©lsy7ux¢|wgzQD |¤V ¤(w}xlgwl2E|)l% ll¤lx d}uDuDE0z}!|} ||0y7l©Dlcxhu llx gd}Qh0| l0(wgyzx|)©l lxl ` Vw g||s 0yP}Dl (u |x)©l8 x s x l % s x s u l h|ê l d$ 0y 'V0 l) l$x | 7y ll x
94 CA´ LCULO INTEGRAL P(x) xl w©l l ©l x 07ª e¦ ª s l | ul | l 8 s 0 w e| xw ! l Q(x) n Ak (x − a)k% Vw v y) kg=}1¤0 d y s l§ l y % (x − a)n j 0V ©l x % | l | y l ¤ }{z Úc l 8 l d Q(x) a n m Bk x + CkQ%§ V(Cqxk)©k s=¤k12w}| ¡l
(I7y lxlD©l)ux−xy d8zxα|0l )zw2|lD|z+ysg¤{z{z¦
βl©'l©x 2©x k8vxDkw h zll l)|gDx 7ys8lzl }l w ¤xDl }yhRx0l 7 |2 lxY7w}y|e2 }u | 7y l x ys l | βl l x 0 wl Ûey 0g l }|¤0 7y ©l x ms l t|x Vα¦ Ri } } h g l©Rx ©l x x©{z ê Ak Bk y x d D 0} | ©l xRx } |dx = x−a x−a +C dx −1 x n =W 1 (x − a)n = (n − 1) (x − a)n−1 + C } | ) y uAx + B (x − α)2 + β2 dx = õ A(x − α) Aα+B (x − α)2 + β2 + (x − α)2 + β2 ö dx = A j (x − α)2 + β2m + A α + B õ x−α +C, 2 xβlDuw β ö¥ d 0 l | l x − α = βt l |4 | }| 7y lDu 8z } u 8 l| y7l s ) )q¦g} 8 á 8 | Ax + B @H ¡ (x − α)2 + β2k k dx = A(x − α) Aα+B ¡ (x − α)2 + β2k k + ¡ (x − α)2 + β2k k dx = −A +©β8 2k g¦ k}− 1x+−(αA α + B) In (t) , = | l¥ l 2(kx − w1)©l xY¡y (xBs −y α x)2l s t In(t) = dt ,}| 7y lDu 8¤ wl § x(lt2¥ + βl)xY2y)gw k !8 | y)l x Ejem p7 Vlo% s .| l 1.– x x2 + 5x − 3 dx 2 Il©=x 0(Ix +x ¤ 10 )z(|x2 − 4) x2 + 5x − 3 ABC (x + 1)(x2 − 4) = x + 1 + x − 2 + x + 2 .
3.1 PRIMITIVAS 95 w 8 | 0 |Q(x) = (x − a) qa(x) s D l| l)Yx yl © x s xly l | l qa(a) =W 0 P(x) = A + R(x) , Q y (gx})¤©8 | x−a qa(x) § w s %V (x − a) qa(x) P(x) = A+ (x − a) R(x) , qs aI() x) qa(x) w lDu s x =a A = P(a) ; wl)xYqy 7 a (a) Dx {z l 4| | h g } j q−1(x) = x2 − 4 s 0l y s f )l x w y ll } | | } |I = 7 + + 11 − − 9 x 1 x 2 x+2 . 3 12 42.– I= 7x4 + 25x3 + 6x2 − 34x + 23 dx 1 ji§ qa(a) =W 0 vD l| l©Yx yl Q(x)© x s xwl 8 y| l | l D V|(x − 1)2(x + 2)3 m> = (x − a)m qa(x) P(x) = A0 A1 Am−1 R(x) (x − a)m−1 x−a qa(x) Qs (xw) − a)m s+ (x + ...+ + , g}¤©8| wDl u y %V (x − a)m qPla(| (xyx)Vj)I7 |g=}} | Al 0a}+ s A l1l (x − a) +...+ Am−1 (x − a)m−1 + j (x − a)m−1ml| l y f ¤ wl } x A|x kmqxa −(|wx1) x 0 l eÛ 0 l | 7y l©x l )l x )7} }} w § } V0 l V l l ¿ 0'0 l | 7y l P(x) l | y V7|x8lI wy ¦l|©l 'syy 7§`l f aVl ¦
¦ 8y 0lztl yw
l Rl E l©dz Dx0sl}d}|z0 yxl¤0|=yl ©l x l w lDu 's )l xYy l l©x )qa7 (x}})}2 y l| | ls = )| 8z ¤l ww } d 0 1 a z `l| l©Þ Yx )y 7x A¨ } k! ©l x ß l 7
8| xw l©x ª
¦) £ 5t xD{z s l | l l l h g } s ¤P(x)=qa(x) j A0 + A1 (x − a) + . . . + Am−1 (x − a)m−1 m + j (x − a)m−1m qa(x) j A0 + A1 (x − a) + . . . + Am−1 (x − a)m−1 m ¤7x4 + 25x3 + 6x2 − 34x + 23 (x + 2)3 (A0 + A1(x − 1))
96 CA´ LCULO INTEGRAL x = 1 =⇒ A0 = 1 ¤28x3 + 75x2 + 12x − 34 A1 (x + 2)3 + 3(x + 2)2 A0h§ s % V R y xh = 1s =⇒ A1 = 2 ¤7x4 + 25x3 + 6x2 − 34x + 23 (x − 1)2 j B0 + B1(x + 2) + B2(x + 2)2m x = −2 =⇒ B0 = 3 ¤28x3 + 75x2 + 12x − 34 (x − 1)2 j B1 + 2 B2(x + 2)m j + 2(x − 1) B0 + B1(x + 2)m x = −2 =⇒ B1 = 4 ¤84x2 + 150x + 12 2(x − 1) B1 + (x − 1)2 2B2 + 2B0 + 2(x − 1) B1 x = −2 =⇒ B2 = 5 } e| ) } l | y7ls I = −1 + 2 | } | x−1 x−1 x+2ê − 3 − 4 +5 2 (x + 2)2 x+23.– uI8= x (x 2 x3 − 3x + 6 1)2 dx | −l)x20x+h %5)(x x¤0− z | x3 − 3x + 6 A B Mx + N 1 wl)2} ¨g=8} ¢ u l)(g¦x| y −)l ¤01V)2ê s + + x2 − ,l | d y D 0 g| (xz |)l2Dx −s l)0 x2w x y+ 5l )|l (xl −8z l¤ − y ) 1| A=1 2§ x + 5 0 7y l l ©l xxw B = x3 − 3x + 6 1 x3 − x2 − x + 1 x +14%d0 .w(– xlDx¤u02gkc−llz I| 8B2=xl | −+Hx ¤x5)l −1d)xY(1Dy xD +x −| 211l))xx |2}(x|−x} hy2©l (gu−x0 ©l)2−2xx©l 1x+)l025)=2y +% ( x)l 2Al −y/Du(}2xD xj x−+−2 1a5m)l)(+xIw)C−©l ¢1cl)© 2¥ = x2 − 2x + . 5 d4 l w }{z Ú|e s )l x l 0 )vh£ a }|dx 1 õ (x − 1)3 − · 1 ) yu õ 2x· + 1 x3 − 1 = 6 x3 − 1 ö 3ö +C @· 3H }|dx 1 x2 + x6 + 1 = · · 3x+1 3 3x+1 4 yu x2 − yu y u + 1 ) x + 1 ) · 3) + 1 ) · 3) + C . (2x − (2x +gÞ x ©l} hx g w)l©l©!ß q l } |l}y l©ll©
xu l) 5© ©5 xl d|3xD w0x}% y| |)l©l¦Ix )w Yw|4 § 6l d 0lzy l |gDlz yV¨ 2 8 }|e7y lx 7 u|eVw yl yª|
h l7y¨ |l l 60}g| 0x¤{z gw§)l xx u0
I xYh) y g x l q x d D w0z yz ª| £
3.1 PRIMITIVAS 97M|¤ w|l ´ely| tl}og|x ldx u8¤l©oh|qxwg(ldxjPe)H(¥x§(H)Uxlm)ee§ Qr(§l mxQ7(M2)x(i)stx (7yDe)xl )s¨|ls ¢0)04à|x©lÃÜhxl%H eÛ l(0x}x |yl)l |
¦=y78©lw x!}l© l}ux||l 7yy| jl8l0Qs7yl l©lye|(x7)x0)¤l©,|}wx Q|eyl ¤ V©w(l xl )xum !§l § M5l u ( x l )w l©x l |V P(x) z=|x¤Q0w |e(x ) /w x H|%( x}) £P(x) = h(x) + m(x) dx .Q(x) dx H(x) M(x)}l©Ò|±0Yxy7yrDxDluqeu nDDu pDwlµgÛ8 p70 q%©l lÄÒx |2p4 ¤yj©l§x³ x©r ©l±©l}ÒÄ pwuhD0±)(dr7xl )|y²2)´ § q% r©mvx¬% §(x xl ))dRx0 lD2 }0|x}7yy0llD|87yu |l©l 7xu8 2lhsV| §|Iy0l)RA| }w%z| |7yd lDu l|8l| ¤ y w }Ûele| ©¤ dw0V)l )z | s l©ux$l x7w8
³©8 0©r| ©±zÄÒ |p V l l h e s dx −x 2 dx(x3 − 1)2 = 3 (x3 − 1) − 3 x3 − 1}| ) y ux6 + 2x4 + 2x3 − 3x2 − 2 x3 + 1 dx = x2(x2 + 1) + x + x+C x3(x2 + 1)2F w ug| n0 c$¥ zil|o8n2 ed 0s}xRt|weri|) g ox¤ un© o}8 | m 7y Dl¦g´eu }t r85x ic ual ls|¦
(l)z I7 }I¤)))¦g l ¤ wl D |h
' l y7l | l | w |eq}| y7Dl u 8 l w e| xl | x, 0 x x) dx ,R( | l R(x, y) ©l x w |e| lw |g0 z | d §0} e| 8 yHl) x x gw x| I x }
}|) 7)¦g} l©xRDx lds |)l x x ll 0 8s w |uDBD l | yl P(x, y)/Q(x, y) P(x, y) Q(x, y(a) l l xl 0 8D |e8}} )Ú h0| l © 8 g¦ } 7y u p x = t s %V l w 8 x ly l | l 2qxl | D xx 1 − t2 = 2t x = 1 + t2 , dx = 2 dt 1 + t2 , 1 + t2 . ¿wl) l| xl R z x
l | y 8Y x x x © 8g¦ } xRx u w l | 7y ©l x ê(b)jjj R(−x, y) = −R(x, y) : ))) ))) qqqg¦gg¦¦ }}} y70xul | x x=t ; R(x, −y) = −R(x, y) x=t ; R(−x, −y) = R(x, y) : x = t. :
98 CA´ LCULO INTEGRAL wl u y 84| x % x ª¦g} } l©x I ) q w |g0 z | s gw ©l x R R(x, y) = R(x, y) − R(−x, y) + R(x, y) + R(−x, −y) + R(−x, y) − R(−x, −y) xvx | 2s l©x 2 2§ } vx x w ¨) | l y
¦8 l | 7y lgs l xy % xjX s%jX §EjX } Ejercicios. 8¤ w ©5 x }| yl©u 0 ) l©x ê @ H x l |+xx+l | 0 x x · 2 } u · 2|| 2((0220 xx x ))x xHx+++· 2 x xl l| dx = · | x +C 1 (2x) 4 2− · 2 x 0 x @ −2 − C x +(2xxl )| · 2 u 4x · x l 0 4x = 4 · 2 + xl 2− (1 + 0 x1 xl | 0xdx = 1 x − 1 } u (1 + 0 x x) + 1 } u (1 − 0 x x) + C x) + 4 2(1 x) 4Funciones irracionales | y7Dl u 8 l©$x z l| yd% 0} e| )R Vs l x§1.– l)xw e| q w |g0 p xl , xk n+ 1| /1d1l¦
,) M.7.¤ ).=¦g,x©l!nx k0 /d(kdq 1 dx | } ni/di P ± R l ¥ 8 ll s ,.. ., © 8 g¦ } x = dk tM )Ejercicio. 8} w ¤) 5q} | 7y Dl u 8 · 1 · dx 31+x− 41+x2.– | y7Dl u 8 ©l x l y % s |R p x, yn1/d1, . . . , ynk/dk dx l } x ni/di P ±ss = ax + b DV| s8§ l©x w |I w g| 0 z |q q d0} |e8 l k+1
) 7 )g¦ )l x y l ¥ c8x +l l ad−bc =W 0 R d © 8¦g} s l | M = 2 D (d1, . . . , dk ) y = tMEjercicio. 8} w ¤) 5q } | 7y Dl u 8 1 1 dx õ x+1 3 x−1ö x+13d.–0}|e 8| y7Dl u 8 l©xcl y % R p x, c ax2 + bx + c dx | l R ©l x5w e| h w |g0 z | l x
)7 )g¦ l©x q(a) C%a©m y b io sl tyrigg| oVn2 o} m ´etl rxiDlcuow s|. h| u g7 ls f l wd u 2 ) x l cx o u mw pl |ley tl aê un cuadra-do ax2 + bx + c = õ x+ b 2 4ac − b2 +, a 2a ö 4a
3.1 PRIMITIVAS 99`§ l | y |g )l Dx s 0 | l ©8 g¦ } x + b = u q}| 7y Dl u 8 x l } l
2 w | l xy % x 2a j £ cR p u, A2 − u2 dx : u = xl | t. q A j £ cR p u, A2 + u2 dx : u = y u t, h g¦ l | ¥ q A u = A t. j £ cR p u, u2 − A2 dx : u= DAx t , h g¦ l | u = A ¥ t. wl xl¤ d0 } |e8 } }Ú)) E| 0 |4} x q x }| ¤© x © 8 g¦ } E| y © dh0 }4|e}8| }7yÚ8l ) | l©0l xYy yd 0x E x u©le8zru .g¦ V}©ã' lDx s ww©l lq l!y 8 lq g| ¦g0 lz| z }|| y ul©lu ©l0lz xDy 7ds ¤ l©|(¤ bw)hl C Í aVm0ó 73 bu ï il osIA| s´H³ çd7©l e²XxYªçÒsy ¤ 0 )¼ Uñ 2Ç1 j £ a > s c· ax2 + bx + c = s x a + t . 0 j £ c > s c· ax2 + bx + c = tx s c . 0 j £ § s § sax2 + bx + c = a(x − α)(x − β) a<0 c<0 c ax2 + bx + c = (x − α) t .(yl0j ªcd}0¢ %à)}|eh¡ ¦Dx8N )7Is}£ o§0¢8Ú àh8t V Dilc¦ãzizWa| ¨sEô l©3 dxlïd)e0j©ll}7Yx3yme|ñ} )x2)´ze}'}tÚ)w |o)|yd©l l©ou©xYl©07y)xdwgl l©e| xDy}ys¨A© %Vlb7e Yuxly.}xv|el8 8x|Vzs}wgs¨©l wl |eV7yz82l
xl lz | ylxYxy¤wy0l))s |w|) |Iy|l0z8z)Vy)l | {z lwyw ¤lD00xu sêg¦ s0©lx hVx g¦gE e l0l0y lthsI¦y0 l©l |x£ }| l j t £ l©xvw `| %} }|} l| s l | y g| l©x P(x) x · P(x) cdx = q(x) ax2 + bx + c + λ · dx ,w ax2 + bx + c l Vu 8w| )}y7|elax 2 +xY!x )y h|bytlx0l 7+u|}V Yx c)y 8 |} 0 l eÛq0(xll )|| 7y ©l©lx x!Rx ¤ w wP|l l r ql%(xu)} |e|s xql } }| r y7lDllDy0ulP|(8x })0e| sl8l§| l ©l | vx ª7y¦
w©l 8x|e| u} | D Dl |qyxYl y y wl¦s|e§ | x | l l m Ûe0 λ jt £ y % · dx (x − α)m ax2 + bx + cxl l© w l ) y %!8 | 7y l 7 } V 5 D | l © 8 g¦ } x − α = 1 t
100 CA´ LCULO INTEGRALjt £ l m P r q } | 7y Dl u 8 dxxl (ax2 + bx +l c)(2m+1)/2 Abel d 0 } e| 8 }Ú)0| cambio de ct = p ax2 + bx + c V = 2ax + b . · q 2 ax2 + bx + cj t £ u¥ l m P r q } | 7y lDu 8 x c dx (x2 + λ)m αx2 + βx l d0}e| 8 }Ú)0 | l ) ) q¦g} c αx2 + β = t ¿ áÓ¤ u}| 7y Dl u 8 dx c(x2 + λ)m αx2 + βx l d0 }e| 8} )Ú 0 | l )) q ¦g} lt ¦ l p c αx2 + β V = t qE© 8 jqer¦gc}icxRiol©Yxsy.ew ) ¤ w x ¤ê)u xqx u w l | y7©l x } | y lDu 8 ©l Dx s 4 z qx l2w | l } x 7 V ¦I8 | c1.– x2 + 4x + 3 dx 2.– · x dx x2 + 4x + 5 3.– · x dx 4.– · dx (x − 2) −x2 + 4x − 3 −x2 + 6x − 5 5.– · x + 3 dx 6.– · x3 − x − 1 dx 4x2 + 4x − 3 x2 + 2x + 2 7.– dx 8.– dx · dx (2x2 − x + 2)7/2 (x − 1)3 x2 − 2x − 1 9.– x+1 dx 10.– (16 − 9x2)3/2 x6 dx · (x2 + 1)2 x2 + 2
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