Bài tập toán cao cấp tập 3 part 1 pot

NGUY ˆ E ˜ N THUY ’ THANH B ` AI T ˆ A . P TO ´ AN CAO C ˆ A ´ P Tˆa . p3 Ph´ep t´ınh t´ıch phˆan. L´y thuyˆe ´ t chuˆo ˜ i. Phu . o . ng tr`ınh vi phˆan NH ` AXU ˆ A ´ TBA ’ NDA . IHO . CQU ˆ O ´ C GIA H ` AN ˆ O . I Mu . clu . c 10 T´ıch phˆan bˆa ´ tdi . nh 4 10.1 C´ac phu . o . ng ph´ap t´ınh t´ıch phˆan . . . . . . . . . . . . 4 10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa ´ td i . nh 4 10.1.2 Phu . o . ng ph´ap d ˆo ’ ibiˆe ´ n 12 10.1.3 Phu . o . ng ph´ap t´ıch phˆan t`u . ng phˆa ` n 21 10.2 C´ac l´o . p h`am kha ’ t´ıch trong l´o . p c´ac h`am so . cˆa ´ p 30 10.2.1 T´ıch phˆan c´ac h`am h˜u . uty ’ 30 10.2.2 T´ıch phˆan mˆo . tsˆo ´ h`am vˆo ty ’ d o . n gia ’ n 37 10.2.3 T´ıch phˆan c´ac h`am lu . o . . ng gi´ac . . . . . . . . . . 48 11 T´ıch phˆan x´ac d i . nh Riemann 57 11.1 H`am kha ’ t´ıch Riemann v`a t´ıch phˆan x´ac d i . nh . . . . . 58 11.1.1 D - i . nhngh˜ıa 58 11.1.2 D - iˆe ` ukiˆe . nd ˆe ’ h`am kha ’ t´ıch 59 11.1.3 C´ac t´ınh chˆa ´ tco . ba ’ ncu ’ a t´ıch phˆan x´ac d i . nh . . 59 11.2 Phu . o . ng ph´ap t´ınh t´ıch phˆan x´ac d i . nh 61 11.3 Mˆo . tsˆo ´ ´u . ng du . ng cu ’ a t´ıch phˆan x´ac d i . nh 78 11.3.1 Diˆe . n t´ıch h`ınh ph˘a ’ ng v`a thˆe ’ t´ıch vˆa . tthˆe ’ 78 11.3.2 T´ınh d ˆo . d`ai cung v`a diˆe . n t´ıch m˘a . t tr`on xoay . . 89 11.4 T´ıch phˆan suy rˆo . ng 98 11.4.1 T´ıch phˆan suy rˆo . ng cˆa . n vˆo ha . n 98 11.4.2 T´ıch phˆan suy rˆo . ng cu ’ a h`am khˆong bi . ch˘a . n . . 107 2MU . CLU . C 12 T´ıch phˆan h`am nhiˆe ` ubiˆe ´ n 117 12.1 T´ıch phˆan 2-l´o . p 118 12.1.1 Tru . `o . ng ho . . pmiˆe ` nch˜u . nhˆa . t 118 12.1.2 Tru . `o . ng ho . . pmiˆe ` ncong 118 12.1.3 Mˆo . t v`ai ´u . ng du . ng trong h`ınh ho . c 121 12.2 T´ıch phˆan 3-l´o . p 133 12.2.1 Tru . `o . ng ho . . pmiˆe ` n h`ınh hˆo . p 133 12.2.2 Tru . `o . ng ho . . pmiˆe ` ncong 134 12.2.3 136 12.2.4 Nhˆa . nx´etchung 136 12.3 T´ıch phˆan d u . `o . ng 144 12.3.1 C´ac d i . nh ngh˜ıa co . ba ’ n 144 12.3.2 T´ınh t´ıch phˆan d u . `o . ng 146 12.4 T´ıch phˆan m˘a . t 158 12.4.1 C´ac d i . nh ngh˜ıa co . ba ’ n 158 12.4.2 Phu . o . ng ph´ap t´ınh t´ıch phˆan m˘a . t 160 12.4.3 Cˆong th´u . c Gauss-Ostrogradski . . . . . . . . . 162 12.4.4 Cˆong th´u . cStokes 162 13 L´y thuyˆe ´ t chuˆo ˜ i 177 13.1 Chuˆo ˜ isˆo ´ du . o . ng 178 13.1.1 C´ac d i . nh ngh˜ıa co . ba ’ n 178 13.1.2 Chuˆo ˜ isˆo ´ du . o . ng 179 13.2 Chuˆo ˜ ihˆo . itu . tuyˆe . td ˆo ´ iv`ahˆo . itu . khˆong tuyˆe . tdˆo ´ i . . . 191 13.2.1 C´ac d i . nh ngh˜ıa co . ba ’ n 191 13.2.2 Chuˆo ˜ id an dˆa ´ u v`a dˆa ´ uhiˆe . u Leibnitz . . . . . . 192 13.3 Chuˆo ˜ il˜uy th`u . a 199 13.3.1 C´ac d i . nh ngh˜ıa co . ba ’ n 199 13.3.2 D - iˆe ` ukiˆe . n khai triˆe ’ nv`aphu . o . ng ph´ap khai triˆe ’ n 201 13.4 Chuˆo ˜ iFourier 211 13.4.1 C´ac d i . nh ngh˜ıa co . ba ’ n 211 MU . CLU . C3 13.4.2 Dˆa ´ uhiˆe . udu ’ vˆe ` su . . hˆo . itu . cu ’ a chuˆo ˜ i Fourier . . . 212 14 Phu . o . ng tr`ınh vi phˆan 224 14.1 Phu . o . ng tr`ınh vi phˆan cˆa ´ p1 225 14.1.1 Phu . o . ng tr`ınh t´ach biˆe ´ n 226 14.1.2 Phu . o . ng tr`ınh d ˘a ’ ng cˆa ´ p 231 14.1.3 Phu . o . ng tr`ınh tuyˆe ´ nt´ınh 237 14.1.4 Phu . o . ng tr`ınh Bernoulli . . . . . . . . . . . . . . 244 14.1.5 Phu . o . ng tr`ınh vi phˆan to`an phˆa ` n 247 14.1.6 Phu . o . ng tr`ınh Lagrange v`a phu . o . ng tr`ınh Clairaut255 14.2 Phu . o . ng tr`ınh vi phˆan cˆa ´ pcao 259 14.2.1 C´ac phu . o . ng tr`ınh cho ph´ep ha . thˆa ´ pcˆa ´ p 260 14.2.2 Phu . o . ng tr`ınh vi phˆan tuyˆe ´ n t´ınh cˆa ´ p2v´o . ihˆe . sˆo ´ h˘a ` ng 264 14.2.3 Phu . o . ng tr`ınh vi phˆan tuyˆe ´ n t´ınh thuˆa ` n nhˆa ´ t cˆa ´ p n n n (ptvptn cˆa ´ p n n n)v´o . ihˆe . sˆo ´ h˘a ` ng . . . . . . 273 14.3 Hˆe . phu . o . ng tr`ınh vi phˆan tuyˆe ´ n t´ınh cˆa ´ p1v´o . ihˆe . sˆo ´ h˘a ` ng290 15 Kh´ai niˆe . mvˆe ` phu . o . ng tr`ınh vi phˆan d a . o h`am riˆeng 304 15.1 Phu . o . ng tr`ınh vi phˆan cˆa ´ p 1 tuyˆe ´ n t´ınh d ˆo ´ iv´o . i c´ac d a . o h`amriˆeng 306 15.2 Gia ’ iphu . o . ng tr`ınh d a . o h`am riˆeng cˆa ´ p2do . n gia ’ n nhˆa ´ t 310 15.3 C´ac phu . o . ng tr`ınh vˆa . tl´y to´an co . ba ’ n 313 15.3.1 Phu . o . ng tr`ınh truyˆe ` n s´ong . . . . . . . . . . . . 314 15.3.2 Phu . o . ng tr`ınh truyˆe ` n nhiˆe . t 317 15.3.3 Phu . o . ng tr`ınh Laplace . . . . . . . . . . . . . . 320 T`ai liˆe . u tham kha ’ o 327 Chu . o . ng 10 T´ıch phˆan bˆa ´ td i . nh 10.1 C´ac phu . o . ng ph´ap t´ınh t´ıch phˆan . . . . . . 4 10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa ´ td i . nh 4 10.1.2 Phu . o . ng ph´ap dˆo ’ ibiˆe ´ n 12 10.1.3 Phu . o . ng ph´ap t´ıch phˆan t`u . ng phˆa ` n 21 10.2 C´ac l´o . p h`am kha ’ t´ıch trong l´o . p c´ac h`am so . cˆa ´ p 30 10.2.1 T´ıch phˆan c´ac h`am h˜u . uty ’ 30 10.2.2 T´ıch phˆan mˆo . tsˆo ´ h`am vˆo ty ’ d o . n gia ’ n 37 10.2.3 T´ıch phˆan c´ac h`am lu . o . . ng gi´ac . . . . . . . 48 10.1 C´ac phu . o . ng ph´ap t´ınh t´ıch phˆan 10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa ´ tdi . nh D - i . nh ngh˜ıa 10.1.1. H`am F (x)du . o . . cgo . i l`a nguyˆen h`am cu ’ a h`am f(x) trˆen khoa ’ ng n`ao d ´onˆe ´ u F (x)liˆen tu . c trˆen khoa ’ ng d´o v`a kha ’ vi 10.1. C´ac phu . o . ng ph´ap t´ınh t´ıch phˆan 5 ta . imˆo ˜ idiˆe ’ m trong cu ’ a khoa ’ ng v`a F  (x)=f(x). D - i . nh l ´y 10.1.1. (vˆe ` su . . tˆo ` nta . i nguyˆen h`am) Mo . i h`am liˆen tu . ctrˆen d oa . n [a, b] dˆe ` u c´o nguyˆen h`am trˆen khoa ’ ng (a, b). D - i . nh l´y 10.1.2. C´ac nguyˆen h`am bˆa ´ tk`ycu ’ a c`ung mˆo . t h`am l`a chı ’ kh´ac nhau bo . ’ imˆo . th˘a ` ng sˆo ´ cˆo . ng. Kh´ac v´o . id a . o h`am, nguyˆen h`am cu ’ a h`am so . cˆa ´ p khˆong pha ’ i bao gi`o . c˜ung l`a h`am so . cˆa ´ p. Ch˘a ’ ng ha . n, nguyˆen h`am cu ’ a c´ac h`am e −x 2 , cos(x 2 ), sin(x 2 ), 1 lnx , cos x x , sin x x , l`a nh˜u . ng h`am khˆong so . cˆa ´ p. D - i . nh ngh˜ıa 10.1.2. Tˆa . pho . . pmo . i nguyˆen h`am cu ’ a h`am f(x) trˆen khoa ’ ng (a, b)d u . o . . cgo . i l`a t´ıch phˆan bˆa ´ td i . nh cu ’ a h`am f(x) trˆen khoa ’ ng (a, b)v`ad u . o . . ck´yhiˆe . ul`a  f(x)dx. Nˆe ´ u F (x) l`a mˆo . t trong c´ac nguyˆen h`am cu ’ a h`am f(x) trˆen khoa ’ ng (a, b) th`ı theo d i . nh l´y 10.1.2  f(x)dx = F (x)+C, C ∈ R trong d ´o C l`a h˘a ` ng sˆo ´ t`uy ´y v`a d˘a ’ ng th´u . ccˆa ` nhiˆe ’ ul`ad ˘a ’ ng th ´u . cgi˜u . a hai tˆa . pho . . p. C´ac t´ınh chˆa ´ tco . ba ’ ncu ’ a t´ıch phˆan bˆa ´ td i . nh: 1) d   f(x)dx  = f(x)dx. 2)   f(x)dx   = f(x). 3)  df (x)=  f  (x)dx = f(x)+C. T`u . d i . nh ngh˜ıa t´ıch phˆan bˆa ´ tdi . nh r ´ut ra ba ’ ng c´ac t´ıch phˆan co . ba ’ n (thu . `o . ng d u . o . . cgo . i l`a t´ıch phˆan ba ’ ng) sau d ˆay: 6Chu . o . ng 10. T´ıch phˆan bˆa ´ td i . nh I.  0.dx = C. II.  1dx = x + C. III.  x α dx = x α+1 α +1 + C, α = −1 IV.  dx x =ln|x|+ C, x =0. V.  a x dx = a x lna + C (0 <a= 1);  e x dx = e x + C. VI.  sin xdx = −cos x + C. VII.  cos xdx = sinx + C. VIII.  dx cos 2 x =tgx + C, x = π 2 + nπ, n ∈ Z. IX.  dx sin 2 x = −cotgx + C, x = nπ, n ∈ Z. X.  dx √ 1 −x 2 =    arc sin x + C, −arc cos x + C −1 <x<1. XI.  dx 1+x 2 =    arctgx + C, −arccotgx + C. XII.  dx √ x 2 ± 1 =ln|x + √ x 2 ± 1|+ C (trong tru . `o . ng ho . . pdˆa ´ utr`u . th`ı x<−1 ho˘a . c x>1). XIII.  dx 1 −x 2 = 1 2 ln    1+x 1 −x    + C, |x|=1. C´ac quy t˘a ´ c t´ınh t´ıch phˆan bˆa ´ td i . nh: 10.1. C´ac phu . o . ng ph´ap t´ınh t´ıch phˆan 7 1)  kf(x)dx = k  f(x)dx, k =0. 2)  [f(x) ± g(x)]dx =  f(x)dx ±  g(x)dx. 3) Nˆe ´ u  f(x)dx = F (x)+C v`a u = ϕ(x) kha ’ vi liˆen tu . cth`ı  f(u)du = F (u)+C. C ´ AC V ´ IDU . V´ı du . 1. Ch´u . ng minh r˘a ` ng h`am y = signx c´o nguyˆen h`am trˆen khoa ’ ng bˆa ´ tk`y khˆong ch´u . ad iˆe ’ m x = 0 v`a khˆong c´o nguyˆen h`am trˆen mo . i khoa ’ ng ch´u . ad iˆe ’ m x =0. Gia ’ i. 1) Trˆen khoa ’ ng bˆa ´ t k`y khˆong ch´u . ad iˆe ’ m x = 0 h`am y = signx l`a h˘a ` ng sˆo ´ . Ch˘a ’ ng ha . nv´o . imo . i khoa ’ ng (a, b), 0 <a<bta c´o signx =1 v`a do d ´omo . i nguyˆen h`am cu ’ a n´o trˆen (a, b) c´o da . ng F (x)=x + C, C ∈ R. 2) Ta x´et khoa ’ ng (a, b)m`aa<0 <b. Trˆen khoa ’ ng (a, 0) mo . i nguyˆen h`am cu ’ a signx c´o da . ng F(x)=−x+C 1 c`on trˆen khoa ’ ng (0,b) nguyˆen h`am c´o da . ng F (x)=x + C 2 .V´o . imo . i c´ach cho . nh˘a ` ng sˆo ´ C 1 v`a C 2 ta thu du . o . . c h`am [trˆen (a, b)] khˆong c´o d a . o h`am ta . idiˆe ’ m x =0. Nˆe ´ u ta cho . n C = C 1 = C 2 th`ı thu du . o . . c h`am liˆen tu . c y = |x| + C nhu . ng khˆong kha ’ vi ta . id iˆe ’ m x =0. T`u . d ´o, theo di . nh ngh˜ıa 1 h`am signx khˆong c´o nguyˆen h`am trˆen (a, b), a<0 <b.  V´ı du . 2. T`ım nguyˆen h`am cu ’ a h`am f(x)=e |x| trˆen to`an tru . csˆo ´ . Gia ’ i. V´o . i x  0 ta c´o e |x| = e x v`a do d´o trong miˆe ` n x>0mˆo . t trong c´ac nguyˆen h`am l`a e x . Khi x<0 ta c´o e |x| = e −x v`a do vˆa . y trong miˆe ` n x<0mˆo . t trong c´ac nguyˆen h`am l`a −e −x + C v´o . ih˘a ` ng sˆo ´ C bˆa ´ tk`y. Theo d i . nh ngh˜ıa, nguyˆen h`am cu ’ a h`am e |x| pha ’ i liˆen tu . cnˆenn´o 8Chu . o . ng 10. T´ıch phˆan bˆa ´ td i . nh pha ’ i tho ’ am˜andiˆe ` ukiˆe . n lim x→0+0 e x = lim x→0−0 (−e −x + C) t´u . cl`a1=−1+C ⇒ C =2. Nhu . vˆa . y F (x)=          e x nˆe ´ u x>0, 1nˆe ´ u x =0, −e −x +2 nˆe ´ u x<0 l`a h`am liˆen tu . c trˆen to`an tru . csˆo ´ .Tach´u . ng minh r˘a ` ng F(x) l`a nguyˆen h`am cu ’ a h`am e |x| trˆen to`an tru . csˆo ´ . Thˆa . tvˆa . y, v´o . i x>0 ta c´o F  (x)=e x = e |x| ,v´o . i x<0th`ıF  (x)=e −x = e |x| . Ta c`on cˆa ` n pha ’ i ch´u . ng minh r˘a ` ng F  (0) = e 0 = 1. Ta c´o F  + (0) = lim x→0+0 F (x) −F (0) x = lim x→0+0 e x − 1 x =1, F  − (0) = lim x→0−0 F (x) −F (0) x = lim x→0−0 −e −x +2− 1 x =1. Nhu . vˆa . y F  + (0) = F  − (0) = F  (0) = 1 = e |x| .T`u . d ´o c ´o t h ˆe ’ viˆe ´ t:  e |x| dx = F(x)+C =    e x + C, x < 0 −e −x +2+C, x < 0.  V´ı du . 3. T`ım nguyˆen h`am c´o d ˆo ` thi . qua diˆe ’ m(−2,2) dˆo ´ iv´o . i h`am f(x)= 1 x , x ∈ (−∞, 0). Gia ’ i. V`ı (ln|x|)  = 1 x nˆen ln|x| l`a mˆo . t trong c´ac nguyˆen h`am cu ’ a h`am f(x)= 1 x . Do vˆa . y, nguyˆen h`am cu ’ a f l`a h`am F (x)=ln|x|+ C, C ∈ R.H˘a ` ng sˆo ´ C d u . o . . cx´acd i . nh t`u . d iˆe ` ukiˆe . n F (−2) = 2, t´u . cl`a ln2 + C =2⇒ C =2−ln2. Nhu . vˆa . y F (x)=ln|x|+2− ln2 = ln    x 2    +2.  10.1. C´ac phu . o . ng ph´ap t´ınh t´ıch phˆan 9 V´ı du . 4. T´ınh c´ac t´ıch phˆan sau dˆay: 1)  2 x+1 −5 x−1 10 x dx, 2)  2x +3 3x +2 dx. Gia ’ i. 1) Ta c´o I =   2 2 x 10 x − 5 x 5 ·10 x  dx =   2  1 5  x − 1 5  1 2  x  dx =2   1 5  x dx − 1 5   1 2  x dx =2  1 5  x ln  1 5  − 1 5  1 2  x ln 1 2 + C = − 2 5 x ln5 + 1 5 ·2 x ln2 + C. 2) I =  2  x + 3 2  3  x + 2 3  dx = 2 3  x + 2 3  + 5 6   x + 2 3  dx = 2 3 x + 5 9 ln    x + 2 3    + C.  V´ı du . 5. T´ınh c´ac t´ıch phˆan sau d ˆay: 1)  tg 2 xdx, 2)  1 + cos 2 x 1 + cos 2x dx, 3)  √ 1 −sin 2xdx. Gia ’ i. 1)  tg 2 xdx =  sin 2 x cos 2 x dx =  1 −cos 2 x cos 2 x dx =  dx cos 2 x −  dx =tgx − x + C. [...]... a a o o a u(x) = x2, dv = sin 3xdx 1 Khi d´ du = 2xdx, v = − cos 3x v` a o 3 1 1 2 2 I = − x2 cos 3x + x cos 3xdx = − x2 cos 3x + I1 3 3 3 3 ` ınh a o Ta cˆn t´ I1 D˘t u = x, dv = cos 3xdx Khi d´ du = 1dx, a 1 u o v = sin 3x T` d´ 3 1 1 2 1 x sin 3x − sin 3xdx I = − x2 cos 3x + 3 3 3 3 2 2 1 cos 3x + C = − x2 cos 3x + x sin 3x + 3 9 27 ´ Nhˆn x´t Nˆu d˘t u = sin 3x, dv = x2dx th` lˆn t´ phˆn t`.ng... + 1 e2x dx (DS ex + ln|ex − 1| ) ex − 1 √ 2 1 + lnx dx (DS (1 + lnx )3) 4 x 3 √ 1 + lnx dx 5 xlnx √ √ (DS 2 1 + lnx − ln|lnx| + 2ln| 1 + lnx − 1| ) 3 6 7 8 9 dx x x (DS −x − 2e− 2 + 2ln (1 + e 2 )) x +e √ √ arctg x dx √ (DS (arctg x)2) x 1+ x √ 2 e3x + e2xdx (DS (ex + 1 )3/ 2 ) 3 ex/2 e2x 2 +2x 1 (2x + 1) dx dx ex − 1 10 √ 11 e2xdx √ e4x + 1 12 2x dx √ 1 − 4x 13 (DS 1 2x2+2x 1 e ) 2 √ (DS 2arctg ex − 1) ... o 1 a ınh ıch a 10 .1 C´c phu.o.ng ph´p t´ t´ phˆn a 7 8 9 11 3x 22x − 1 √ dx 2x 2 22 x + 2− 2 ) (DS ln2 3 lnx 1 (DS √ arctg √ ) 2 2 dx x(2 + ln2 x) √ 3 ln2 x dx x 3 5 /3 ln x) 5 (DS 10 ex + e2x dx 1 − ex 11 ex dx 1 + ex 12 x sin2 dx 2 (DS 13 cotg2 xdx (DS −x − cotgx) 14 √ π 1 + sin 2xdx, x ∈ 0, 2 15 ecos x sin xdx 16 ex cos ex dx (DS sin ex) 17 1 dx 1 + cos x x (DS tg ) 2 18 dx sin x + cos x 19 1. .. − 1 ln (1 + x2)) 2 √ √ (DS (1 + x)arctg x − x) x3 x x4 − 1 arctgx − + ) 4 12 4 10 x3 arctgxdx 11 (arctgx)2xdx (DS 12 (arc sin x)2dx √ (DS x(arc sin x)2 + 2arc sin x 1 − x2 − 2x) 13 arc sin x √ dx x +1 √ √ (DS 2 x + 1arc sin x + 4 1 − x) 14 arc sin x dx x2 √ 1 + 1 − x2 arc sin x − ln ) (DS − x x 15 xarctgx √ dx 1 + x2 (DS (DS x2 + 1 1 (arctgx)2 − xarctgx + ln (1 + x2)) 2 2 √ √ 1 + x2arcrgx − ln(x + 1 +... Chu.o.ng 10 T´ phˆn bˆt dinh 28 1 x2x dx 2 x2 e−x dx 3 x3 e−x dx 4 (x3 + x)e5xdx 5 arc sin xdx 6 xarc sin xdx 7 x2 arc sin 2xdx 8 arctgxdx 9 √ arctg xdx (DS 2x (x ln 2 − 1) ) ln2 2 (DS −x2e−x − 2xe−x − 2e−x ) 1 2 (DS − (x2 + 1) e−x ) 2 2 (DS 31 1 5x 3 3 2 31 e x − x + x− ) 5 5 25 12 5 (DS xarc sin x + (DS √ 1 − x2 ) 1 1 √ (2x2 − 1) arc sin x + x 1 − x2) 4 4 (DS 2x2 + 1 √ x3 arc sin 2x + 1 − 4x2) 3 36 (DS... t´ phˆn bang v` t´ c´c t´ch phˆn d 1 e a ınh a ı a o dx x4 − 1 1 (DS 1 1 x 1 ln − arctgx) 4 x +1 2 1 1 + 2x2 dx (DS arctgx − ) x2 (1 + x2 ) x √ √ √ x2 + 1 + 1 − x2 √ dx (DS arc sin x + ln|x + 1 + x2|) 1 − x4 √ √ √ √ x2 + 1 − 1 − x2 √ dx (DS ln|x + x2 − 1| − ln|x + x2 + 1| ) x4 − 1 √ 1 x4 + x−4 + 2 dx (DS ln|x| − 4 ) 3 x 4x 2 3 4 5 23x − 1 dx ex − 1 6 (DS e2x + ex + 1) 2 ’ ´ ´ a u o ’ o e Dˆ cho gon,... 2arctg ex − 1) (DS (DS √ 1 ln(e2x + e4x + 1) ) 2 arc sin 2x ) ln2 √ √ dx √ (DS 2[ x + 1 − ln (1 + x + 1) ]) 1+ x +1 ’ ˜ Chı dˆ n D˘t x + 1 = t2 a a 14 x +1 √ dx x x−2 15 √ dx ax + b + m √ √ (DS 2 x − 2 + 2arctg (DS x−2 ) 2 √ 2 √ ax + b − mln| ax + b + m| ) a a ınh ıch a 10 .1 C´c phu.o.ng ph´p t´ t´ phˆn a 16 dx √ √ 3 3 x( x − 1) 17 dx (1 − x2 )3/ 2 √ √ (DS 3 3 x + 3ln| 3 x − 1| ) (DS tg(arc sin x)) ’... 3x) 2 1 + 9x 6 √ 1 1 x + arctg2x dx (DS ln (1 + 4x2) + arctg3/22x) 1 + 4x2 8 3 arc sin x − arc cos x √ dx 1 − x2 26 x + arc sin3 2x √ dx 1 − 4x2 27 x + arc cos3/2 x √ dx 1 − x2 28 x|x|dx 29 (DS 1 (arc sin2 x + arc cos2 x)) 2 (DS − 25 1 1 1 − 4x2 + arc sin4 2x) 4 8 (2x − 3) |x − 2|dx 30 (DS √ 2 (DS − 1 − x2 − arc cos5/2 x) 5 |x |3 ) 3  − 2 x3 + 7 x2 − 6x + C, x < 2  3 2 ) (DS F (x) = 2 3 7 2  x − x... = xdx Khi d´ du = o dx 2 3 1 · √ , v = x 2 Do d´ o 1+ x 2 x 3 √ 1 2 3 x dx I = x 2 arctg x − 3 3 1+ x √ 1 1 2 3 1 dx = x 2 arctg x − 3 3 1+ x √ 2 3 1 = x 2 arctg x − (x − ln |1 + x|) + C 3 3 V´ du 2 T´ I = arc cos2 xdx ı ınh ’ ’ ’ o Giai Gia su u = arc cos2 x, dv = dx Khi d´ 2arc cos x du = − √ dx, v = x 1 − x2 Theo (10 .4*) ta c´ o xarc cos x √ dx 1 − x2 ’ ’ ´ ’ a Dˆ t´ t´ phˆn o vˆ phai d˘ng th´.c... Chı dˆ n D˘t x = a cos 2t 33 x − 1 dx x + 1 x2 1 (DS arc cos − x 1 ˜ ’ a Chı dˆ n D˘t x = a t √ dx √ 34 (DS 2arc sin x) x − x2 √ x2 − 1 ) x a ınh ıch a 10 .1 C´c phu.o.ng ph´p t´ t´ phˆn a 21 ˜ ’ a Chı dˆ n D˘t x = sin2 t a √ √ √ 1 + x2 + 1 x2 + 1 dx (DS x2 + 1 − ln ) 35 x x 36 x3dx √ 2 − x2 (9 − x2)2 dx x6 37 38 (DS − x2dx √ x2 − a2 (DS x2 √ 4√ 2 − x2 − 2 − x2) 3 3 (DS − (9 − x2 )5 ) 45x5 √ . ho . c 12 1 12 .2 T´ıch phˆan 3- l´o . p 13 3 12 .2 .1 Tru . `o . ng ho . . pmiˆe ` n h`ınh hˆo . p 13 3 12 .2.2 Tru . `o . ng ho . . pmiˆe ` ncong 13 4 12 .2 .3 13 6 12 .2.4 Nhˆa . nx´etchung 13 6 12 .3 T´ıch. . 19 2 13 . 3 Chuˆo ˜ il˜uy th`u . a 19 9 13 . 3 .1 C´ac d i . nh ngh˜ıa co . ba ’ n 19 9 13 . 3.2 D - iˆe ` ukiˆe . n khai triˆe ’ nv`aphu . o . ng ph´ap khai triˆe ’ n 2 01 13 . 4 Chuˆo ˜ iFourier 211 13 . 4 .1. phˆan h`am nhiˆe ` ubiˆe ´ n 11 7 12 .1 T´ıch phˆan 2-l´o . p 11 8 12 .1. 1 Tru . `o . ng ho . . pmiˆe ` nch˜u . nhˆa . t 11 8 12 .1. 2 Tru . `o . ng ho . . pmiˆe ` ncong 11 8 12 .1 .3 Mˆo . t v`ai ´u . ng du . ng