Tich Phan Phuongphapnhaytanglau

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http://ebooktoan.com/forum 1 NGUYN HM V TCH PHN BI 1. BI TPSDNGCNG THC NGUYN HM, TCH PHN I.Bng cng thc nguyn hm m rng ( )1111ax bax b dx c,ao+o + | |+ = + o= |o+ \ .} ( ) ( )1cos ax bdx sinax ba+ = +} + c 1 dxln ax b cax b a= + ++} + c ( ) ( )1sinax bdx cos ax b ca+ = + +} 1ax b ax be dx e ca+ += +} ( ) ( )1tg ax bdx ln cos ax b ca+ = + +} 1ax b ax bm dx m ca ln m+ += +} ( ) ( )1cotg ax bdx ln sinax b ca+ = + +} 2 21 dx xarctg ca a a x= ++} ( )( )21 dxcotg ax b ca sin ax b= + ++} 2 212dx a xln ca a x a x+= + } ( )( )21 dxtg ax b ca cos ax b = + ++} ( )2 22 2dxln x x a cx a= + + ++} 2 2x xarcsin dx x arcsin a x ca a= + +} 2 2dx xarcsin caa x= +} 2 2x xarccos dx x arccos a x ca a= +} 2 21 dx xarccos ca ax x a= +}( )2 22x x aarctg dx xarctg lna x ca a= + +} 2 22 21 dx a x aln ca xx x a+ += ++} ( )2 22x x aarccotg dx xarccotg ln a x ca a= + + +} ( ) ( )blnax bdx x lnax b x ca| |+ = + + + |\ .} ( )12dx ax bln tg csinax b a+= ++} 2 2 22 22 2x a x a xa x dx arcsin ca = + +} ( )12dx ax bln tg csinax b a+= ++} ( )2 2axaxe asinbx bcosbxe sinbxdx ca b= ++} ( )2 2axaxe acosbx bsinbxe cosbx dx ca b+= ++} http://ebooktoan.com/forum 2 II. NHNG CH KHI S DNG CNG THC KHNG C TRONG SGK 12 CccngthccmttrongII.mkhngctrongSGK12khisdngphichngminhli bngcchtrnhbydidngb.Cnhiucchchngminhbnhngcchngin nht l chng minh bng cch ly o hm 1. V d 1: Chng minh: 2 2dx 1 x aln c2a x ax a= ++}; 2 2dx 1 a xln c2a a xa x+= +} Chng minh: 2 2dx 1 1 1 1 dx dx 1 x adx ln c2a x a x a 2a x a x a 2a x ax a | | | |= = = + || + + + \ . \ .} } } } ( )2 2dx 1 1 1 1 dx d a x 1 a xdx ln c2a a x a x 2a a x a x 2a a xa x| | + | |= + = = + ||+ + \ . \ .} } } } 2. V d 2: Chng minh rng: ( )2 22 2dxln x x ax a= + ++} + c Chng minh: Ly o hm ta c: ( )( )2 22 22 21 x aln x x a cx x a''+ + (+ + + = + + =2 22 2 2 2 2 2 2 2 2 21 x 1 x x a 11x x a x a x x a x a x a+ + | |= + = = |+ + + + + + +\ . 3. V d 3: Chng minh rng:2 2dx 1u caa x= ++} (vi xtg ua=) t xtg ua= , ( )u ,2 2t te ( )( )2 22 2d a tg udx 1 1du u ca aa xa 1 tg u= = = +++} } } 4. V d 4: Chng minh rng: 2 2dxu ca x= +} (vi xsin ua= , a > 0) t xsin ua= ,ue ,2 2t t( ( ( )( )2 22 2dx d a sin udu u ca xa 1 sin u= = = +} } } Bnh lun: Trc nm 2001, SGK12 c cho s dng cng thc nguyn hm2 2dx 1 xarctg ca aa x = ++}v 2 2dx xarcsin caa x= +}(a>0)nhngsaukhnggingbtc nc no trn th gii, h li cm khng cho s dng khi nim hm ngc arctg x, arcsin x. Cch trnh by trn khc phc lnh cm ny. III. CC DNG TCH PHN N GIN III.1. CC K NNG C BN: 1. Biu din lu tha dng chnh tc:http://ebooktoan.com/forum 3 =1nnx x ;= =m mn n k m mn nkx x ; x x= =1nnnn1 1x ; xx x ;=mnn m1xx; =mnkn k m1xx 2. Bin i vi phn:dx d(x 1) d(x 2) d(x p) adx d(ax 1) d(ax 2) d(ax p) ( ) ( )x p 1x 1 x 2dx d d da a aa | | = = = = |\ .LIII.2. CC BI TP MU MINH HO 1. 3dx1xx }( )321 1 1dx 1 dx1 1xx xx x + | |= = + + + | \ .} } = ( )( )2 3 211 11 dx ln 11 3 2d xx x x x x x cx + + + = + + + +} } 2.( )14 7 dx = 4 7 7 4 7 dx4x x x x ++ ( + } } ( ) ( ) ( ) ( ) ( )3 5 3 12 2 2 21 1 2 24 7 7 4 7 4 7 4 7 7 4 716 16 5 3x x d x x x c ((= + + + = + + + ( } 3. ( )( )( )172 2 2d 2d 12 5 22 5xxIxx= =++} }1 10arctg510x c| |= + |\ . 4. ( )( )( )xdx 1 2 1 1 1 1 22 lnln 2 5ln 2 5ln 2 2 + 5 2 2 5 2 52 2 5x xxx x xx xdd c| |= = = + |+ + \ . +} } } 5. ( ) ( )53 2 3coscos 1 sin 1 sin cos cos sin dx1 sinxdx x x dx x x x xx (= + = + } } } ( ) ( ) ( )3 42 3sin cos1 sin sin cos cos sin3 4x xx d x xd x x c = = +} } III.3. CC BI TP DNH CHO BN C T GII ( ) ( ) ( )( )1x 1 x 2 x 3 x 4J dxx x+ + + +=} ; 27x 3J dx2x 5=+} ; 233x 7x 5J dxx 2 +=} ( )3 2 2 24 5 6102x 5x 7x 10 4x 9x 10 2x 3x 9J dx ; J dx ; J dxx 1 2x 1x 1 + + += = = } } } ( ) ( )3 2 3 27 815 30x 3x 4x 9 2x 5x 11x 4J dx ; J dxx 2 x 1 + + += = +} } ( ) ( ) ( ) ( ) ( )( )} } } + = + = + = dx 1 x 2 5 x 3 x J ; dx 2 x 5 1 x J ; dx 1 x 3 x J3321115 2103 1009 http://ebooktoan.com/forum 4 ( )( )( )( )2432 4 55 912 13 1447x 3x 5J 2x 3 . x 1 dx ; J dx ; J x . 2x 3 dx2x 1 += + = = ++} } } ( )9 315 16 174 2 2105x x xJ dx ; J dx ; J dxx x 1 x x 12 3x= = =+ } } } ( ) ( )( )( ) ( )( )18 19 202 2 2 2dx dx dxJ ; J ; Jx 2 x 5x 2 x 6 x 2 x 3= = = ++ + +} } } ( )( ) ( ) ( ) ( ) ( )21 22 232 2 2 2 2 2x dx dx dxJ ; J ; Jx 3 x 7 3x 7 x 2 2x 5 x 3= = = + + + } } } ln 2 ln 2 ln 2 ln 22x xx24 25 26 27xx x1 0 0 0dx e dx 1 eJ ; J ; J e 1dx ; J dx1 ee 1 e 1= = = + =+ +} } } } ( ) ( )2 2x x1 1 1 1x28 29 30 31x 2x 2x x 3x0 0 0 01 e dx 1 ee dx dxJ ; J ; J ; J dx1 e 1 e e e e+ += = = =+ + +} } } } ln 2 ln 4 1 e3x32 33 34 35x 3 x x x0 0 0 1dx dx e dx 1 ln xJ ; J ; J ; J dxx e e 4e 1 e+ += = = = +} } } } ( )3 1 165 2 5 3 3 236 37 380 0 0J x 1 x dx ; J x 1 x dx ; J x 1 x dx = + = = } } } ( )2x1 1 1 12x x39 40 41 42x x x x0 0 0 02 1 dxdx dxJ ; J ; J ; J e 1 e dx4 3 4 2 4 += = = = ++ +} } } } BI 2. TCH PHN CC HM S C MU S CHA TAM THC BC 2 A. CNG THC S DNG V K NNG BIN I 1.= ++} 2 2du 1 uarctg ca a u a4.= +}du2 u cu 2. = ++ } 2 2du 1 u aln c2a u a u a 5. ( )= + >}2 2du uarcsin c a 0aa u 3. += + } 2 2du 1 a uln c2a a u a u 6.= + +}22duln u u p cu p K nng bin i tam thc bc 2: 1. ( | |+ + = + ( |\ . ( 2222b b 4acax bx c a x2a 4a 2.( ) + + = + 22 2ax bx c mx n pB. CC DNG TCH PHNhttp://ebooktoan.com/forum 5 I. Dng 1: } 2dxA=ax + bx + c 1. Phng php: ( )+= = ++ ++ +} } 2 22dx dx 1 mx narctg cmp p ax bx cmx n p

( )+ = = ++ + + ++ } } 2 22mx n p dx dx 1ln c2mp mx n p ax bx cmx n p 2. Cc bi tp mu minh ha ( )( )( ) ( )12 2 22d d 1 d 2 2 1 2 2 3ln2 4 8 1 4 3 2 2 32 2 32 2 3x x x xA cx x xxx+ + = = = = ++ + + ++ + } } } 3. Cc bi tp dnh cho bn c t gii: 12dxA3x 4x 2= } ;2 32 2dx dxA ; A ;4x 6x 1 5x 8x 6= = + + +} } 2 1 14 5 62 2 21 0 0dx dx dxA ; A ; A7x 4x 3 6 3x 2x 4x 6x 3= = = + + +} } } II. Dng 2:( )} 2mx + nB = dxax + bx + c 1. Phng php:( )( )( )+ + += =+ + + +} } 2 2m mb2ax b nmx n2a 2aB dx dxax bx c ax bx c == ( )+ +| |+ |+ + \ .}22d ax bx cm mbn A2a 2a ax bx c = | |+ + + |\ .2m mbln ax bx c n A2a 2a Cch 2: Phng php h s bt nh (s dng khi mu c nghim) Nu mu c nghim kp 0x x =tc l 2 20( ) ax bx c ax x + + = th ta gi s: ( )+= + + +2 200mx nxx x ax bx cx x Quy ng v phi v ng nht h s hai v tm o, |. Vi o, | va tm ta c: ( ) +=+ +} 2mx nB dxax bx c =ln +00x x cx xNumuc2nghimphnbit 1 2, x x : 21 2( )( ) ax bx c ax x x x + + = thtagis += + + +21 2mx nxx x x x ax bx c http://ebooktoan.com/forum 6 Quy ng v phi v ng nht h s hai v tm o, |. Vi o, | va tm ta c: ( )dx+=+ +} 2mx nBax bx c =ln ln + +1 2x x x x c 2. Cc bi tp mu minh ha: }122x + 3B = dx9x 6x + 1 ( )( )2 2 21 1118 61 18 6 d 11 d9 3d9 3 9 6 1 9 6 1 9 6 1xx x xxx x x x x x += = + + + +} } } ( ) ( )( )( )22 21 9 6 1 11 3 1 2 11ln 3 19 9 9 9 3 1 9 6 13 1d x x d xx cx x xx + = + = + +} } 3. Cc bi tp dnh cho bn c t gii: ( ) ( ) ( )1 2 32 2 27 3x dx 3x 4 dx 2 7x dxB ; B ; B4x 6x 1 2x 7x 9 5x 8x 4 = = = + } } }; III. Dng 3: }2dxC =ax + bx + c 1. Phng php:B : ln22duu u k cu k = + + ++} Bin i nguyn hm v 1 trong 2 dng sau: ( )( ) ( )22 2dx dx 1ln C mx n mx n k cmax bx cmx n k= = = + + + + ++ ++ +} } ( )( )2 22dx dx 1arcsin 0mx nC pm pax bx cp mx n+= = = >+ + +} } 2. Cc bi tp mu minh ha: ( )( )232 2d 1 d 55 45ln4 162 4454 10 55416x xC x x cx xx= = = + + } } 3. Cc bi tp dnh cho bn c t gii: 1 2 32 2 2dx dx dxC ; C ; C3x 8x 1 7 8x 10x 5 12x 4 2 x= = = + } } } IV. Dng 4: ( )}2mx + n dxD =ax + bx + c 1. Phng php:( )2 22 dxdx2 2ax bm mbDa aax bx c ax bx c+= + + + +} }= ( )222 2d ax bx cm mbCa aax bx c+ + + +} 2. Cc bi tp mu minh ha:http://ebooktoan.com/forum 7 D1 =( ) ( )1 1 12 2 20 0 04 d 2 d d24 5 4 5 4 5x x x x xx x x x x x+ += ++ + + + + +} } } ( )( )( )( )1 1 1 22 22 200 01 d 4 5 d2 4 5 2ln 2 4 524 52 1x x xx x x x xx xx+ += + = + + + + + + ++ ++ +} } ( ) ( )3 1010 5 2ln 3 10 2ln 2 5 10 5 2ln2 5+= + + + = ++ 3. Cc bi tp dnh cho bn c t gii: ( ) ( ) ( )1 2 32 2 25 4x dx 3x 7 dx 8x 11 dxD ; D ; D3x 2x 1 2x 5x 1 9 6x 4x + = = = + } } } V. Dng 5: ( )}2dxE =px + q ax + bx + c 1. Phng php: t 21 dt 1 1dx ; px q p x qt p t t | |+ = = = |\ .. Khi : ( )22 2 22dt pt dx dtEpx q ax bx c t t1 a 1 b 1q q ct t p t p= = = + + + o +| + | | | | + + ||\ . \ .} } } 2. Cc bi tp mu minh ha: ( )}3122dxE =x - 1 x - 2x + 2. t 22 111 131 ;2dxx ttx tx xt tdtt= = + = = = = = Khi : ( )( ) ( )1 2 3212 22 1dt t dxE1 x-1 x 2x 2t 1 t 12 2t tt= = ++ + +} } ( )11221 21 2dt 1 5 2 2 2ln t t 1 ln 1 2 ln ln21 5t 1+ += = + + = + =++} 3. Cc bi tp dnh cho bn c t gii: ( ) ( ) ( )2 3 31 2 32 2 21 2 2dx dx dxE ; E ; E2x 3 x 3x 1 3x 4 2x 3x 7 x 1 x 1= = =+ + + + +} } } VI. Dng 6: ( )( )}2mx + n dxF =px + q ax + bx + c http://ebooktoan.com/forum 8 1. Phng php: ( )( )( )( )2 2dxdxmqmpx q nmx n p pFpx q ax bx c px q ax bx c| |+ + |+\ .= =+ + + + + +} } ( )2 2dx dx mq mq m mF n C n Ep pp pax bx c px q ax bx c| | | |= + = + ||\ . \ .+ + + + +} } 2. Cc bi tp mu minh ha: ( )( )11202 3 d1 2 2x xFx x x+= =+ + +}( )1 12 20 0dx dx2 2I Jx 2x 2 x 1 x 2x 2+ = ++ + + + +} } 120dx2 2Ix x=+ +}( )( ) ( )1 12020dx 2 5ln 1 1 1 ln1 21 1x xx+= = + + + + =++ +} ( )1201 2 2dxJx x x=+ + +}. t 20 111112dxx tx txtdtt= = = =+ = =. Khi : ( ) ( )1 2 12 122 21 21 1 2dt t dt 2 2 2J ln t t 1 ln1 51 t 11 11 2 1 2t tt += = = + + =++ + +} } F1 = 2I + J = ( )( ) ( )2 5 2 2 2 2 9 4 52ln ln ln1 2 1 51 2 1 5+ + ++ =+ ++ + ( )( )( )( )3 2225 12 12 22 1 4 3xdxx x x+ +=+ } }-3 222-2x + 3 dxF =2x + 1 -x - 4x - 3

( )3 2 3 22 22 21 dx 5 dx 1 5I J2 2 2 2x 4x 3 2x 1 x 4x 3 = + = + + } } 3 2224 3dxIx x= }( )( )3 23 2222dxarcsin x 261 x 2t= = + = +} ( )3 2222 1 4 3dxJx x x=+ }. t 21231 13 12 12 222x ttx t x x ;t tdtdxt = = = = + = == ( ) ( )( ) ( )1 2 1 322 21 3 1 21 3 1 32 21 2 1 2dt 2 t dtJ1 5 t 6 t 11 1 11 2 1 34 t tt1 dt 1 5 t 3 1 2 1arcs in arcs in ar csin2 3 45 5 53 2t5 5 = = + | |= = = |\ . +} }} http://ebooktoan.com/forum 9 Vy ( ) 255 1 2 1F I J arcsin arcsin2 2 12 2 3 4t= + = + 3. Cc bi tp dnh cho bn c t gii: ( )( )( )( )( )( )1 1 11 2 32 2 20 0 04x 7 dx 6 7x dx 7 9x dxF ; F ; F8 5x 3x 4x 2 2x 5 x x 4 4x 3 2x x 1+ = = = + + + + ++} } } VII. Dng 7: ( )}2 2xdxG =ax + b cx +