一元函数的不定积分

����� ���������� � ��� §3.1 ��������������������ff §3.1.1 fiffiflffi�ffi ffi!ffi" #ffi$ffi%ffi&ffi'ffi(ffi) * +-,/.ffi0ffi1ffi2ffi3ffi4ffi5ffi687ffi98: v(t), ; 4ffi5ffi6ffi2ffi3ffi<ffi= ) +ffi4ffi5ffi6ffi2ffi3ffi<ffi=ffi: s = s(t), s(0) = 0. >ffi?ffi@ffiA s′(t) = v(t). B/CffiDFE/GffiHffiIffiJ ; %ffi&8KffiL8M8N s(t), O s′(t) = v(t), PffiQffiR s(0) = 0. SUTVEWG 6ffiXffiY-Z/[ D]\ffiN8^ 6ffi_ffi9 $ffiD @8?ffi`ffiaffib ;ffic 6ffid82ffie )gf ? D + f(x) h-iffij I k Affilffim D ;-iffij/k KffiL8M8N F (x), O dF (x) dx = F ′(x) = f(x), (x ∈ I). nffio 1 + f(x) h-iffij I Affilffim )qpffir i8jsk KFL8M8N F (x) QffiR F ′(x) = f(x) (x ∈ I), tffiu F (x) ? f(x) 6 %ffi&ffivffiMffiNffi) wUx DgpUr F (x) ? f(x) 6 vUMUNUD t8yffizffi{ffi| N c, F (x) + c } ? f(x) 6 v MffiNffi)q~-B l Y 2.3.5 6ffiffi€ K . f(x) 6 zffi{ffi &ffivffiMffiNffi‚ffiƒ : | N8D…„8† {F (x) + c | c ∈ R} @ffi? f(x) 6ffi‡ffiˆ vffiMffiNffi‰ffiŁ 6ffi‹8Œ ) nUo 2 + F (x) ? f(x) 6 %U&UvUMffiN8D tffiu ‹8Œ {F (x) + c | c ∈ R} : f(x) 6ffi lffiŽffi DqffiŁ ∫ f(x)dx = F (x) + c. 1 h ffi‘ ∫ f(x)dx ’ D “ ∫ ” “ Žffi ‘ffiD f(x) “/” Ž MffiNffiD f(x)dx “/” Žffi• –ffi— D x “ Žffiffi˜ffi™ ) %U&Uš |U› ` 6UYU€ E/G @U? DœUžffiŸ 6 MUN A vffiMUN�  hffi¡ %ffi¢U£ffi¤ffi¥U¦ §-¨/© pffir f(x) h-iffij I ªffi« D tffih I ’ f(x) A vffiMffiNffi) pffir F (x) ? f(x) 6 vffiMffiNffiD t B lffim K . dF (x) = f(x)dx. ¬ Cffi\ffi­ — k $ffiD A ∫ dF (x) = ∫ f(x)dx = F (x) + c ® d ∫ f(x)dx = d(F (x) + c) = dF (x) = f(x)dx. „ffi† Žffi “ ∫ ” ® L  “d” ?  &ffi¯ dffi6ffi28e8° ‘8)±BsL  6 ­ —8 ˜8²8³ . Žffi } A ­ —ffi ˜ffi² D } @8?ffi´ 8µ x ?�¶·˜ffi™ffi¸ffi? ’ffij ˜ffi™ D…¹ A ∫ dF (x) = F (x) + c. ºU% 5 ? ¦U» ;U¼ ffi½ffi˜ffi™ffi¾ L  <ffi=U6ffiYffi€ffi¿UÀ D } ? h ŽffiffiÁ = ’/ bffi˜U™ ÃffiÄ 6ffiYffi€ffi¿ffiÀ ) + F (x) ? f(x) 6 vffiMffiNffiD t y = F (x) u : f(x) 6 %ffiÅ Žffi-Æ 1 DqÇ x y8zffi{ c ∈ R, y = F (x) + c (1) } ? f(x) 6 ŽUVÆ 1 )ÉÈffi¥ ? y = F (x) Ê Oy ffií effi6 ­ — ∫ [af(x) + bg(x)]dx = a ∫ f(x)dx + b ∫ g(x)dx. Ý Ûffiî {ffih í e86 Á = ’qï —Fð8ñFò8óF6  & z8{F| N ( Ç a, b 8‡ffi:ffiô Ú ) õ Kffi† Œ Pffië Łffi%ffi& zffi{8| N8) ö ;ffic 6ffi÷ffiøffiùffi—ffiúffiû Á »8D @8³ J yffiÐ 6 Žffi ù8— D…f A 3 ∫ odx = c; ∫ dx√ 1− x2 = arcsinx + c; ∫ xµdx = xµ+1 µ + 1 + c, µ 6= −1; ∫ dx 1 + x2 = arctan x + c;∫ sinxdx = − cos x + c; ∫ axdx = ax ln a + c;∫ cos xdx = sinx + c; ∫ dx x = ln |x|+ c;∫ sec2 xdx = tan x + c; ∫ chxdx = shx + c;∫ csc 2xdx = − cot x + c; ∫ shxdx = chx + c. * 1 ; ∫ ( 3x2 + 4 x ) dx. ü ∫ ( 3x2 + 4 x ) dx = 3 ∫ x2dx + 4 ∫ dx x = 3 · x 3 3 + 4 ln |x|+ c = x3 + 4 ln |x|+ c. * 2 ; ∫ x2 1 + x2 dx. ü ∫ x2 1 + x2 dx = ∫ ( 1− 1 1 + x2 ) dx = ∫ dx− ∫ dx 1 + x2 = x− arctan x + c. * 3 ; ∫ tan2 xdx. ü ∫ tan2 xdx = ∫ (sec2 x− 1)dx = ∫ sec2 xdx− ∫ 1dx = tanx− x + c. §3.2 ý�þ�����ß�� §3.2.1 ���ffiäffiå�� 4 1) � %ffiÄ���� B Œ MffiN 6 L  �ffiD…¤8¥ .�� A dF (ϕ(x)) = F ′(ϕ(x))dϕ(x) = F ′(ϕ(x))ϕ′(x)dx, Žffi ï — @ffi³ J ∫ F ′(ϕ(x))ϕ′(x)dx = F (ϕ(x)) + c. k�� Ç 6 Åffiá��ffiDq¤8¥ @ “ ��� ” Ü %ffi& A�� ;  lffiŽffi 6 l Y ) nffiè 1 + F (u) ? f(u) 6 vffiMffiNffiD u = ϕ(x) A ªffi«ffic Nffi) t∫ f(ϕ(x))ϕ′(x)dx = F (ϕ(x)) + c. (1) ºffi& “ ��� ” Ü » 6 l Y�������� D u : � %ffiÄ����ffiD�� � L  �ffi) * 1 ; ∫ sin3 xdx. ü ∫ sin3 xdx = ∫ sin2 x(− cos x)′dx = − ∫ (1− cos2 x)(cos x)′dx. � u = cosx, t B l Y 1 � ∫ (1− u2)du = u− u 3 3 + c, @ffi³ J ∫ sin3 xdx = cos3 x 3 − cos x + c. pffirffiö (1) ë Ł ∫ f(ϕ(x))ϕ′(x)dx u=ϕ(x) = ∫ f(u)du = F (u) + c u=ϕ(x) = F (ϕ(x)) + c. tffi; Žffi 6 Á = @�ffffi˜8³�fi�fl Ç8) * 2 ; ∫ lnx x dx. 5 ü ∫ lnx x dx u=lnx = ∫ udu = 1 2 u2 + c = 1 2 ln2 x + c. Õ ��ffi ¢U£ ´UÁ 6 Ò ŸffiD P  Ûffi`� ’Uj ˜ffi™ ϕ(x) %U&�! 6�"�# u, $ š ϕ(x) 6 • –ffi—���% D'& Èffi%8&�(�) 6 ! "�# ) ¬ C (1) ¸ Kffi† ë Ł ∫ f(ϕ(x))ϕ′(x)dx = ∫ f(ϕ(x))dϕ(x) = F (ϕ(x)) + c. * 3 ; ∫ x2 √ 1 + x3dx. ü ∫ x2 √ 1 + x3dx = 1 3 ∫ √ 1 + x3d(1 + x3) = 1 3 1 1 + 12 (1 + x3)1+ 1 2 + c = 2 9 (1 + x3) 3 2 + c. 2) ��ffi Ä���� ¦ 1) 6 Á =ffiú Á »ffiDqö x = ϕ(t) ÃffiÞ ∫ f(x)dx 6 ” Žffi• –ffi— ’ D @ffi³ J ∫ f(x)dx = ∫ f(ϕ(t))ϕ′(t)dt. pffir f(ϕ(t))ϕ′(t) A vffiMffiN F (t), P�* x = ϕ(t) A ú MffiN t = ϕ−1(x), t B Žffi 6 ­ —ffi ˜ffi²ffi@ffiA ∫ f(x)dx = ∫ f(ϕ(t))ϕ′(t)dt = F (t) + c = F (ϕ−1(x)) + C. B ú MffiN 6 ;ffic � tffi} [�+80�,�- § F (ϕ−1(x)) ? f(x) 6 %ffi&ffivffiMffiNffi) :�. ` x = ϕ(t) A ú MffiN�/ h D ÛF` ϕ(t) ? %ffi&�0�1�2�3 6 M8NFD±B l Y 2.3.2 K . D Ý ` ϕ′(t) 4 A ôU5 D ϕ′(t) @  ffU˜ ‘ffiD ¬ Î }�5 ? 0�1�2�3 6 )76 k „�8ffiDq¤ffi¥ A 6 n è 2 + f(x) h i j I A l m D x = ϕ(t) h i j J K L D ϕ′(t) 6= 0, x = ϕ(t) ö J 9 Ł I. :Fh J k f(ϕ(t))ϕ′(t) A vFMFN F (t), t f(x) h I k A vFMFN F (ϕ−1(x)). f A ∫ f(x)dx = ∫ f(ϕ(t))ϕ′(t)dt = F (t) + c = F (ϕ−1(x)) + c. (2) hffiÂ�; 2ffie ’ Ý h�5 `8Ú & y l Y 2 ’ f(x) ® x = ϕ(t) „ Û QffiR 6 Åffiá a b - § Dq%�<�=�> ¡ Ý Û�? (2) — „ • –ffi6ffi=�@ ?�Affi@�B e ¡ T @8b . ) * 4 + |x| ≤ a, ; ∫ √ a2 − x2dx. ü � x = a sin t, |t| ≤ pi2 . t t = arcsin xa , >ffi? ∫ √ a2 − x2dx = ∫ a2 cos2 tdt = a2 ∫ 1 + cos 2t 2 dt = a2 2 t + a2 4 ∫ cos 2td2t = a2 2 t + a2 4 sin 2t + c = a2 2 t + a2 2 sin t cos t + c = a2 2 arcsin x a + x 2 √ a2 − x2 + c. * 5 ; ∫ √ 3− 2x− x2dx. ü ∫ √ 3− 2x− x2dx = ∫ √ 4− (x + 1)2dx = 4 ∫ √ 1− ( x + 1 2 )2 d x + 1 2 = 2 arcsin x + 1 2 + x + 1 2 √ 3− 2x− x2 + c. * 6 + a > 0, ; ∫ dx√ x2 + a2 . ü ∫ dx√ x2 + a2 x=a sh t = ∫ (sh t)′ ch t dt = ∫ dt = t + c1 = sh−1 x a + c1 = ln ( x a + √ x2 a2 + 1 ) + c1 = ln(x + √ x2 + a2) + c (c = c1 − ln a). 7 * 7 ; ∫ dx√ x + 1 . ü ∫ dx√ x + 1 x=t2 = ∫ 2tdt t + 1 = ∫ 2dt− 2 ∫ dt t + 1 = 2t− 2 ∫ d(1 + t) 1 + t = 2t− 2 ln(1 + t) + c = 2 √ x− 2 ln(1 +√x) + c. ¬ : ºU£ òUó t = √ x > 0, „U†UD ∫ d(1 + t) 1 + t = ln(1+t)+c, ÎU 5Uë Ł ln |1+t|+c. h�C ˜8™ Ã8Ä Ú D AFÚFÛED�F x = ϕ(t) ?�G %8% y8Ð † ��H�I ?EG�JEKEL f(x) 6 lffim I ) * 8 ; ∫ √ x2 − a2dx (a > 0). ü � x = ach t, B > ch t h (−∞, 0] ® [0,+∞) A  Õ 6 2�3 ² D�MFÝ J N ê ’ %8& i8j )�O N t ≥ 0. ~ ¬ : ch t > 0, „8†8DqÃ8Ä x = ach t Ý J8³ J√ x2 − a2 h-iffij [a,+∞) k 6ffi lffiŽffi )qfffiÇ x ≥ a Ú∫ √ x2 − a2dx x=a ch t= ∫ a2sh2tdt = a2 ∫ ch 2t− 1 2 dt = −a 2 2 t + a2 2 sh tch t + c1 = −a 2 2 ln(x + √ x2 − a2) + x 2 √ x2 − a2 + c. Ç x ≤ −a Ú D'� x = −ach t (t ≥ 0). @ffiA∫ √ x2 − a2dx = −a2 ∫ sh2 tdt = a2 2 t− a 2 2 sh tch t + c1 = a2 2 ln(−x + √ x2 − a2) + x 2 √ x2 − a2 + c2 = −a 2 2 [ ln(−x− √ x2 − a2) ] + x 2 √ x2 − a2 + c. õ�P »ffiD @ffiA ∫ √ x2 − a2dx = −a 2 2 ln |x + √ x2 − a2|+ x 2 √ x2 − a2 + c. * 9 ; ∫ √ x2 − a2 x dx, a > 0. 8 ü � x = a sec t, Ç x ≥ a Ú D 0 ≤ t < pi2 . º Ú ∫ √ x2 − a2 x dx = a ∫ a tan t a sec t · sec t tan tdt = a ∫ (sec2 t− 1)dt = a tan t− at + c = √ x2 − a2 − a arccos a x + c. x ≤ −a Ú D pi2 < t ≤ pi. º Ú √ x2 − a2 = −a tan t, M ∫ √ x2 − a2 x dx = −a ∫ (sec2 t− 1)dt = −a tan t + at + c1 = √ x2 − a2 + a arccos a x + c1 = √ x2 − a2 − a arccos ( −a x ) + c. õ�P »ffiD @ffiA ∫ √ x2 − a2 x dx = √ x2 − a2 − a arccos a|x| + c. * 10 ; ∫ dx x √ a2 + x2 (a > 0). ü x > 0 Ú D ∫ dx x √ a2 + x2 = ∫ dx x2 √ 1 + (a x )2 = −1a ∫ 1√ 1 + (a x )2 dax = −1 a ln  a x + √ 1 + a2 x2  + c = 1 a ln x a + √ a2 + x2 + c. Ç x < 0 Ú D ∫ dx x √ a2 + x2 = ∫ d(−x) (−x)2 √ 1 + (−ax)2 = 1 a ln −x a + √ a2 + x2 + c. õ�P »ffiD @ffiA ∫ dx x √ a2 + x2 = 1 a ln ∣∣∣∣ xa +√a2 + x2 ∣∣∣∣+ c. 9 §3.2.2 å�Qffiäffiå�� + u(x) ® v(x) ¹ffiKffiLffiD t A (u(x)v(x))′ = u′(x)v(x) + u(x)v′(x). wffix Dqpffir u′(x)v(x) ® u(x)v′(x) ‚ffi% A vffiMffiNffiD t�R %8& } A v8M8N8) ¬ C8¤ ¥ @ffiA n è 3 ( TSUŽ  ù — ) + u(x) ® v(x) K L D u′(x)v(x) A v M N D t u(x)v′(x) } A vffiMffiNffiD P A ∫ u(x)v′(x)dx = u(x)v(x) − ∫ u′(x)v(x)dx. * 1 α 6= −1, ; ∫ xα lnxdx. ü xα = ( xα+1 α + 1 ) ′ , M�� v(x) = x α+1 α + 1 , u(x) = lnx, t A ∫ xα lnxdx = xα+1 α + 1 lnx− 1 α + 1 ∫ xα+1(ln x)′dx = xα+1 α + 1 lnx− 1 α + 1 ∫ xαdx = xα+1 α + 1 ( lnx− 1 α + 1 ) + c. U é DqÇ α = 0 Ú D @ffi³ J ∫ lnxdx = x lnx− x + c. B >ffi�SffiŽffi ùffi— } K8† ë Ł ∫ u(x)dv(x) = u(x)v(x) − ∫ v(x)du(x). „ffi† h ŽffiffiÚ D } Kffi† 0�, “ � L  ”. * 2 ; ∫ x2exdx. 10 ü ∫ x2exdx = ∫ x2dex = x2ex − ∫ exdx2 = x2ex − 2 ∫ xexdx = x2ex − 2 ∫ xdex = x2ex − 2xex + 2 ∫ exdx = (x2 − 2x + 2)ex + c. h ' 2 ’ Dqpffirffiö “ � L  ” 6 MffiNffiÄffiŁ x2, @ffi³  Ü œffižffiIffirffi) * 3 ; ∫ x sinxdx. ü ∫ x sinxdx = − ∫ xd cos x = −x cos x + ∫ cos xdx = sinx− x cos x + c. * 4 ; ∫ arctan xdx. ü ∫ arctan xdx = x arctan x− ∫ x 1 + x2 dx = x arctan x− 1 2 ∫ d(1 + x2) 1 + x2 = x arctan x− 1 2 ln(1 + x2) + c. * 5 + a 6= 0, ; I = ∫ eax cos bxdx ® J = ∫ eax sin bxdx. ü I = 1 a ∫ cos bxdeax = 1 a eax cos bx + b a J, J = 1 a ∫ sin bxdeax = 1 a eax sin bx− b a I. ¼ �ffi> I ® J 6 ffi �ffi%�V�W�X <ffi= ‰ffiD @8³ J I = ∫ eax cos bxdx = b sin bx + a cos bx a2 + b2 eax + c, J = ∫ eax sin bxdx = a sin bx− b cos bx a2 + b2 eax + c. 11 * 6 ; In = ∫ lnn xdx, n ≥ 1. ü ¤ffi¥ A�Y ffiùffi— In = x ln n x− ∫ xd lnn x = x lnn x− n ∫ lnn−1 xdx = x lnn x− nIn−1, I0 = x + c. * 7 ; In = ∫ dx (a2 + x2)n , n ≥ 1. ü In = x (a2 + x2)n − ∫ xd 1 (a2 + x2)n = x (a2 + x2)n + 2n ∫ x2 (a2 + x2)n+1 dx = x (a2 + x2)n + 2nIn − 2na2In+1. @ffi³ J Y ffiùffi— I1 = 1 a arctan x a + c, In+1 = 1 2na2 · x (a2 + x2)n + 2n− 1 2na2 In, (n ≥ 1). „ffi† A I2 = ∫ dx (x2 + a2)2 = x 2a2(a2 + x2) + 1 2a3 arctan x a + c. * 8 + a > 0, x > 0, ; ∫ arcsin √ x x + a dx. ü Z t = arcsin √ x x + a , t x = a tan 2 t. >ffi? ∫ arcsin √ x x + a dx = ∫ tdx = tx− ∫ xdt = xt− a ∫ tan2 tdt = xt− a ∫ (sec2 t− 1)dt = xt + at− a tan t + c = (x + a) arcsin √ x x + a −√ax + c. 12 §3.3 []\���������� * §3.3.1 ^�_�`�affiçffi��b�c�d�e P (x) = 0 “ ô�f�gffi— ) + n ≥ 0, an 6= 0, a0, a1, · · · , an ¹ ? ffiNffiD t P (x) = anx n + an−1x n−1 + · · ·+ a1x + a0 u : %ffi& n V f�gffi— D n = ∂oP u :�f�gffi—ffi6 VffiNffiDqš ô | N ? ô V f�gffi— ) nUè 1 ( hjilk�m ) + P (x) ? f�gU— D Q(x) ? š ô�f�g8— D t A�n % f�g — q(x) ® r(x) QffiR P (x) = q(x)Q(x) + r(x), (1) ê-’ r(x) = 0 � ∂or < ∂oQ. o pffir P (x) = 0, t N q(x) = r(x) = 0 fffiKffi) pffir ∂oP < ∂oQ, t N q(x) = 0, r(x) = P (x) fffiKffi) pffir ∂oP ≥ ∂oQ. + P (x) = anx n + an−1x n−1 + · · ·+ a1x + a0 (an 6= 0), Q(x) = bmx m + bm−1x m−1 + · · ·+ b1x + b0 (bm 6= 0, m ≤ n). >ffi? P (x)− an bm xn−mQ(x) = ( an−1 − anbm−1 bm ) xn−1 + · · · = P1(x). f A P (x) = an bm xn−mQ(x) + P1(x), ê-’ P1(x) = 0 � ∂oP1 ≤ n− 1. pffir P1(x) = 0 � ∂oP1 ≤ m − 1, t N r(x) = P1(x) f ³ J (1). pffir ∂oP1 ≥ ∂oQ, t ~8K8†8ö P1(x) ë Ł Q(x) p %�2 g8— ‚ Ž�q k %8&EV8NEr > ∂oP1 6 f�gU— ) › k 8 Á = DqB > ∂oPk ?�s 6 D @ õ ff8³ J Pl(x) = 0 � ∂oPl < ∂ oQ. º Ú D N r(x) = Pl(x) @ffi³ J (1). êffiÂ�t & §-¨ Á = @ffi? y ∂oP = n O � Nffi^ffiH u��ffi) q § q(x) ® r(x) 6 n % ² )qpffir ¸ffiA P (x) = q1(x)Q(x) + r1(x), ∂ or1 < ∂ oQ � r1(x) = 0, 13 t A (q1(x)− q(x))Q(x) + (r1(x)− r(x)) = 0. (2) pffir q1(x)− q(x) 6= 0, t (q1(x) − q(x))Q(x) 6 V8N ≥ ∂oQ. Î r1(x) − r(x) = 0 � r1(x) − r(x) 6 V8N < ∂oQ. ºffiŸ (2) v ñffi K Jffi? ô�f�gffi— )…„8† 5 A q1(x) = q(x). >ffi? Dq~ 5 A r1(x) = r(x). B l Y 1 6 § ¨ Á = K8†8$ Ü D…pFr P (x) ® Q(x) ¹ ? Â�w N f�g8— D t q(x) ® r(x) } ? Â�w N f�gffi— ) Ç (1) ’ r(x) = 0 Ú D u Q(x) $�x P (x). + P (x) ? %U& n(n ≥ 1) V f�gU— DWpffir P (α) = 0, tUu α ? f�gU— P (x) 6 %ffi&�yffi) nUè 2 + P (x) ? n(n ≥ 1) V f�gU— D t α ? P (x) 6 y 6�z  5 ` Åffiá ? x− α $�x P (x). o + P (α) = 0. B l Y 1 K . D A q(x) ® | N r O P (x) = (x− α)q(x) + r. ¦ x = α ÃffiÞ k — @ffi³ J r = 0. ú Á »ffiD t ?�{ffi ¨ w 6 ) nffiè 3 ( |�}�~��€� ) z %ffi& n(n ≥ 1) V f�gffi— ¹�‚�ƒ A %ffi&�yffi) ºffi& l Y ¦ h ˜ MffiN�„ = ’ §-¨ ) ú Ð � l Y 3 @ffi³ J …�† + P (x) ? %ffi& n(n ≥ 1) V f�gffi— D t�5 A P (x) = A(x− α1)n1(x− α2)n2 · · · (x− αs)ns , (3) ê-’ A 6= 0, α1, · · · , αs ?  Õ 6 ffiN ( f P (x)  Õ 6 y ), Î n1 + n2 + · · ·+ ns = n. 14 (3) ’ 6 ni(i = 1, 2, · · · , s) “ y αi 6 › NffiD u αi ? P (x) 6 ni › yffi)/„ffi†ffi% & n V f�gffi— A n &�y ( ? › N í e y 6 &ffiN ). nffiè 4 Â�w N f�gffi—ffi6 ffiN�y�‡�ˆ8Ł y Ü Offi) o + α ? n(n ≥ 1) V Â�w N f�gffi— P (x) = anx n + an−1x n−1 + · · ·+ a1x + a0 6 ffiN�yffi) t A P (α) = anα n + an−1α n−1 + · · ·+ a1α + a0 = 0, >ffi? P (α) = a¯nα¯ n + a¯n−1α¯ n−1 + · · ·+ a¯1α¯ + a¯0 = 0, ‰ ¬ : a0, a1, · · · , an ¹ ?  Nffi)q„ffi† a¯0 = a0, · · · , a¯n = an. M P (α) = anα¯ n + an−1α¯ n−1 + · · ·+ a1α¯ + a0 = P (α¯) = 0. f α¯ } ? P (x) 6 yffi) pffir α, α¯ ? Â�w N f�gffi— P (x) 6 % y ‡�ˆ� ffiN�yffiD t B l Y 2 K . A P (x) = (x− α)(x − α¯)P1(x) = (x2 − (α + α¯)x + αα¯)P1(x). B > x2 − (α + α¯)x + αα¯ ? Â�w N f�g8— DW„ffi† $ P (x) „ ³ 6�ŁU— P1(x) } ?  w N f�gffi— ) ¬ Cffi¤ffi¥8~ A …�† + P (x) ? %ffi& n(n ≥ 1) V Â�w N f�gffi— ) t�5 A P (x) = A(x− α1)r1 · · · (x− αk)rk(x2 + β1x + γ1)s1 · · · (x2 + βlx + γl)sl , (4) ê ’ A 6= 0, αi, βj , γj ¹ ?  N D * β2j−4γj < 0, � Ç j 6= j′ Ú (βj , γj) 6= (βj′ , γj′) (i = 1, · · · , k, j = 1, · · · , l), Î r1 + · · · + rk + 2s1 + · · ·+ 2sl = n. êU (4) @U? (3), Ý  Á ö yUÐffiÑ % y ‡�ˆ� ffiN�y 6  &ffi%�V —�‹ Łffi%ffi& ffi V — x2 + βx + γ, B > ºffi& ffi V — A % y ‡�ˆ� �yffiD…„8†�Œ é — β2 − 4γ < 0. §3.3.2 Qffiåffiåffiç�� C��ffi„ ´ 6�f�gffi— ¹ ? Â�w N f�g8— D  q� ¨ ) 15 + P (x) ® Q(x) ¹ ? š ô�f�gffi— ) t f(x) = P (x) Q(x) “ A Y MffiN�� “  — M8Nffi) pffir ∂oP < ∂oQ, tffiu P (x) Q(x) :�Ž  — ) B l Y 1 K . Dqpffir Q(x) $  x P (x), t  — MffiN P (x) Q(x) K • Łffi%ffi& f�g — p %ffi& Ž  — ‚ ® D…f A P (x) Q(x) = q(x) + r(x) Q(x) (∂or < ∂oQ). B > f�gU—U6 ŽU�{ffi [�+ D„ffi†ffi¤ffi¥UÝ Û� €�Ž  — p��‘ffiŁ�’�2  — ‚ ® † � ’�2  — p� Ž8 6 E/G8) nffiè 5 + P (x) Q(x) ? %ffi& Ž  — D Q(x) ,  ¼ Ł (4) 6 ­ — ) t n %�“ A P (x) Q(x) = k∑ i=1 ( A1 x− α + · · ·+ Ar (x− α)r ) + l∑ j=1 ( B1x + C1 x2 + βx + γ + · · ·+ Bsx + Cs (x2 + βx + γ)s ) . (5) h�� %ffi& ® — £8D α ® r ¹�”ffiT . ¡�• i, f ø Ð�–ffië Ł αi ® ri. Î A1, · · · , Ar ¹�”ffiT . k�• (i). f ø Ð�–ffië Ł A (i) 1 , · · · , A(i)ri . Õ Ÿ ��ffi & ® — £ β — γ ® s ¹ ”�˜ . ¡�• j, B1, · · · , Bs ® C1, · · · , Cs } ¹�”�˜ . k�• (j). (5) “ Ž  — P (x) Q(x) 6 Sffiffi —  ¼ — ) ¤ffi¥�˜ffiT l Y 5 6 §-¨ ) ™ | ��š l w N���› l (5) ’ 6 w N Ai, Bj , Cj ï ) ¦ (5) v ñ ™  D ê �œ @ffi? Q(x)( � Ô ƒffi%ffi&ffiš ôffi6 | N ¬ ( ). Ý `ž ( v�— ð  ñ  ( ’ x Õ V�Ÿ 6 w NffiD @ Kffi† ³ J �ffi> A,B,C 6 %�V�W�X <ffi= ‰ffiDq\ Î ¼ Ü A,B,C. O � Sffiffi — �ffiD @�J ö Ž  —86 Ž8ffi ¼ Ł �  ’�2  —86 Žffi © 1◦ Ik = ∫ dx (x− α)k ; 2◦ Jk = ∫ Bx + C (x2 + βx + γ)k dx, β2 − 4γ < 0. 16 ; Ik �ffi[�+ ) Î Jk = ∫ Bx + C (x2 + βx + γ)k dx = ∫ Bx + C[( x + β2 )2 + 4γ − β 2 4 ]k dx, � 4γ − β2 4 = p 2, x + β2 = t, f A Jk = ∫ B (t− β2 ) + C (t2 + p2)k dt = B 2 ∫ d(t2 + p2) (t2 + p2)k + ∫ C − Bβ2 (t2 + p2)k dt. k —ffið�¡ � %ffi& Žffi�{8 [�+ D Î B 3.2.2 6 ' 7 K . k —ffið�¡ ��ffi & Žffi Kffi†-B Y ffiùffi—ffie Ü ) ¬ CffiD 1◦, 2◦   ’�2  — MffiN 6 Žffi ¹ ? ßffiï MffiN8) §3.3.3 *�¢ * 1 ; Žffi ∫ dx x2 − a2 (a 6= 0). ü 1 x2 − a2 = 1 (x− a)(x + a) = 1 2a ( 1 x− a − 1x + a ) , M ∫ dx x2 − a = 1 2a (∫ dx x− a − ∫ dx x + a ) = 1 2a ln ∣∣∣∣x− ax + a ∣∣∣∣+ c. £ffi�¤ | �ffi6  ¼ — © 1 (u + a)(u + b) = 1 b− a ( 1 u + a − 1 u + b ) , (a 6= b) (6) ® 1 (u + a)(v − a) = 1 u + v ( 1 u + a + 1 v − a ) , (u 6= −v) (7) * 2 ; ∫ dx x3 + 1 . ü + 1 x3 + 1 = 1 (x + 1)(x2 − x + 1) = A x + 1 + Bx + C x2 − x + 1. >ffi?ffiA   A + B = 0; B −A + C = 0; A + C = 1. 17 ¼ ³ A = 13 , B = −13 , C = 23. M ∫ dx x3 + 1 = 1 3 ∫ dx x + 1 − 1 3 ∫ x− 2 x2 − x + 1dx = 1 3 ln |x + 1| − 1 6 ∫ 2x− 1− 3 x2 − x + 1dx = 1 6 ln (x + 1)2 x2 − x + 1 + 1 2 ∫ dx x2 − x + 1 = 1 6 ln (x + 1)2 x2 − x + 1 + 1 2 ∫ dx( x− 12 )2 + 34 = 1 6 ln (x + 1)2 x2 − x + 1 + 1√ 3 arctan 2x− 1√ 3 + c. * 3 ; ∫ x3 + 1 x4 − 3x3 + 3x2 − xdx. ü x4 − 3x3 + 3x2 − x = x(x− 1)3. + x3 + 1 x(x− 1)3 = A x + B x− 1 + C (x− 1)2 + D (x− 1)3 . >ffi? x3 + 1 = A(x− 1)3 + Bx(x− 1)2 + Cx(x− 1) + Dx.  (  ñ x Õ V�Ÿ 6 w NffiD ³   A + B = 1; −3A− 2B + C = 0; 3A + B − C + D = 0; −A = 1. ¼ ³ A = −1, B = 2, C = 1, D = 2. M A∫ x3 + 1 x4 − 3x3 + 3x2 − xdx = − ∫ dx x + 2 ∫ dx x− 1 + ∫ dx (x− 1)2 + 2 ∫ dx (x− 1)3 = ln (x− 1)2 |x| − 1 (x− 1)2 − 1 x− 1 + c = ln (x− 1)2 |x| − x (x− 1)2 + c. O � SUU — � ? ; A Y MffiN 6 Žffi 6ffi÷Uøffi< �ffiD¥‰�¦�¦�§�¨�©�ª 6 í e D „ffi† h K JffiÚ Ð Ç�«�¬ O � D Î�­���® Ï�U é 68< �8) 18 * 4 ; ∫ x2 + 1 x4 + 1 dx. ü x4 + 1 = (x2 −√2x + 1)(x2 +√2x + 1). ¯ � (7) K ³ 1 x4 + 1 = 1 2(x2 + 1) ( 1 x2 −√2x + 1 + 1 x2 + √ 2x + 1 ) , M ∫ x2 + 1 x4 + 1 dx = 1 2 ∫ dx x2 −√2x + 1 + 1 2 ∫ dx x2 + √ 2x + 1 = 1 2 ∫ d(x− √22 ) ( x− √ 2 2 )2 + 12 + 1 2 ∫ d(x + √22 ) ( x + √ 2 2 )2 + 12 = √ 2 2 arctan( √ 2x− 1) + √ 2 2 arctan( √ 2x + 1) + c. * 5 ; ∫ x2 − 1 x4 + 1 dx. ü ∫ x2 − 1 x4 + 1 dx = ∫ 1− 1 x2 x2 + 1 x2 dx = ∫ d(x + 1x)( x + 1x )2 − 2 = ∫ d(x + 1x)( x + 1x − √ 2 )( x + 1x + √ 2 ) = 1 2 √ 2 ∫ d(x + 1x −√2) x + 1x − √ 2 − 1 2 √ 2 ∫ d(x + 1x +√2) x + 1x + √ 2 = 1 2 √ 2 ln x2 −√2x + 1 x2 + √ 2x + 1 + c. kE° 6 ” Ž MFN h ˜ ­ 6 Ú²± D²³�´ ` ; x 6= 0. º ÚFŽFFFé h (−∞, 0) ® (0,+∞) k aUb )µ‰�¶�� ³ J 6 v8MffiN h x = 0 } ? KUL 6 D >ffi? x 6= 0 6�·�¸ ¶ 3 N�¹ ) Žffi ï — h (−∞,+∞) k Ł�Xffi) * 6 ; ∫ dx x(1 + x7) . 19 ü ∫ dx x(1 + x7) = ∫ x6dx x7(1 + x7) = 1 7 (∫ dx7 x7 − ∫ d(x7 + 1) x7 + 1 ) = 1 7 ln x7 x7 + 1 + c. * 7 ; ∫ x4 + 1 x6 + 1 dx. ü ∫ x4 + 1 x6 + 1 dx = ∫ ( x4 − x2 + 1 x6 + 1 + x2 x6 + 1 ) dx = ∫ dx x2 + 1 + 1 3 ∫ dx3 (x3)2 + 1 = arctan x + 1 3 arctan(x3) + c. §3.3.4 º�»�^ è çffi�ffiäffiå + R(u, v) ?E�F> u, v 6 A Y MFNF)qf B  & ¶ ˜±™ u, v ´ ÁFA · V½¼ t 2ffie „ ³ J 6 M8N8D w8x R(u, v) Kffi† ë Ł  & ffi � f�g8—86EŁ )…º8£8¤8¥  € R(cos x, sinx) 6 Žffi E/Gffi) 1) ¾ J ÃffiÄ � t = tan x2 ( ¾ J ÃffiÄ ), t A cos x = 2 cos2 x 2 − 1 = 1− t 2 1 + t2 , sinx = 2 sin x 2 cos x 2 = 2 tan x 2 · cos2 x 2 = 2t 1 + t2 , dx = 2d arctan t = 2 1 + t2 dt. >ffi? ∫ R(cos x, sinx)dx = 2 ∫ R ( 1− t2 1 + t2 , 2t 1 + t2 ) dt 1 + t2 . º ? t 6 A Y MffiNffiD @ Kffi† � S8ffi — � Ž8 ) 20 * 1 ; ∫ dx 5 + 4 sinx . ü � t = tan x2 . t ∫ dx 5 + 4 sinx = 2 ∫ dt 5t2 + 8t + 5 = 2 5 ∫ d(t + 45 ) ( t + 45 )2 + ( 3 5 )2 = 2 3 arctan 5t + 4 3 + c. ê-’ t = tan x2 . ¾ J ÃffiÄ�¦�¦�¿ffi»�©�ª 6 í e D Ð�–�x K J «�¬ O � ) 2) pffir ” Ž MffiN ? tanx 6 A Y MffiN R(tan x), � ˜ffi™ ÃffiÄ t = tanx @ffi³ J ∫ R(tanx)dx = ∫ R(t) 1 + t2 dt. * 2 ; ∫ sin2 x cos x sinx + cos x dx. ü B sin2 x = 1 − 1 1 + tan2 x , cos xsinx + cos x = 1 1 + tanx . M8K8† � ˜8™ Ã8Ä t = tan x. ∫ sin2 x cos x sinx + cos x dx = ∫ t2 (1 + t)(1 + t2)2 dt = 1 4 ∫ ( 1 1 + t − t− 1 1 + t2 + 2t− 2 (1 + t2)2 ) dt = 1 4 ln |1 + t|√ 1 + t2 − 1 + t 4(1 + t2) + c = 1 4 ln | sinx + cos x| − 1 4 cos x(cos x + sinx) + c. * 3 ; ∫ 1− tan x 1 + tan x dx. ü � t = tan x, t∫ 1− tanx 1 + tanx dx = ∫ 1− t 1 + t · dt 1 + t2 = ∫ ( 1 1 + t − t 1 + t2 ) dt = ln |1 + t|√ 1 + t2 + C = ln | cos x + sinx|+ c. 3) ® Ï Ú�± D � Ã8Ä t = cos x � t = sinx @�J ö�À _