Phuong phap tich phan

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  • 1. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt1CHUYN 1TCH PHNA. TM TT KIN THCI. Bng tnh nguyn hm c bn:Bng 1 Bng 2Hm s f(x) H nguyn hm F(x)+C Hm s f(x) H nguyn hm F(x)+Ca ( hng s) ax + Cx11xC(ax b) a11( )1ax bC 1xln x C 1ax b1ln ax b Ca xalnxaCaxexe Cax be 1ax be Ca sinx -cosx + C sin(ax+b)1cos(ax b) Ca cosx sinx + C cos(ax+b)1sin(ax b) Ca 21cos xtanx + C21cos (ax b)1tg(ax b) Ca 21sin x-cotgx + C21sin (ax b)1cot g(ax b) Ca '( )( )u xu xln u(x) C2 21x a1ln2x aCa x atanxln cos x C2 21x a2 2ln x x a Ccotx ln sin x CV d 1: Tm h nguyn hm ca cc hm s sau:1.31( ) cos1f x xx x 2. 22x 5f(x)x 4x 3 V d 2: Tnh cc tch phn: 1.5cos x sin xdx 2.costgxdx x 3.1 ln xdxxII. NH NGHA V CC TNH CHT CA TCH PHN1. nh ngha:

2. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt2Cho hm s y=f(x) lin tc trn a;b . Gi s F(x) l mt nguyn hm ca hm s f(x) th:( ) ( ) ( ) ( )bbaa f x dx F x F b F a ( Cng thc NewTon - Leiptnitz)2. Cc tnh cht ca tch phn: Tnh cht 1: Nu hm s y=f(x) xc nh ti a th : ( ) 0ba f x dx Tnh cht 2: ( ) ( )b aa b f x dx f x dx Tnh cht 3: Nu f(x) = c khng i trn a;b th: ( )ba cdx c b a Tnh cht 4: Nu f(x) lin tc trn a;b v f (x) 0 th ( ) 0ba f x dx Tnh cht 5: Nu hai hm s f(x) v g(x) lin tc trn a;b v f (x) g(x) xa;b th( ) ( )b ba a f x dx g x dx Tnh cht 6: Nu f(x) lin tc trn a;bv m f (x) M ( m,M la hai hang so) th( ) ( ) ( )bam b a f x dx M b a Tnh cht 7: Nu hai hm s f(x) v g(x) lin tc trn a;b th ( ) ( ) ( ) ( )b b ba a a f x g x dx f x dx g x dx Tnh cht 8: Nu hm s f(x) lin tc trn a;b v k l mt hng s th. ( ) . ( )b ba a k f x dx k f x dx Tnh cht 9: Nu hm s f(x) lin tc trn a;b v c l mt hng s th( ) ( ) ( )b c ba a c f x dx f x dx f x dx Tnh cht 10: Tch phn ca hm s trn a;b cho trc khng ph thuc vo bin s ,ngha l : ( ) ( ) ( ) ...b b ba a a f x dx f t dt f u du B. CC PHNG PHP TNH TCH PHNI. Phng php phn tch. 3. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt3* Ni dung: S dng cc php bin i i s kt hp vi cc tnh cht ca tch phn a tch phncn tm v cc tch phn c trong bng nguyn hm sau p dng nh ngha.Cc v d:1) Tnh :161 x dx131(x 1) dx40 sin 2x dx20 cos2x dx20 sin4x dx24 cot2x dx2) Tnh:4 2242sintg xx dx30 ( cosxcos3x + sin4xsin3x) dx36 tg2x dx10 e2x + 1dx3) Tnh :40 | x-2 | dx42 2 x 6x 9 dx34 | x2-4 | dx344 cos2x 1 dxBi 1: Tnh cc tch phn sau:1)130xdx(2x 1) 2)10xdx2x 1 3)10 x 1 xdx 4)1204x 11dxx 5x 6 5)1202x 5dxx 4x 4 6)3 320xdxx 2x 1 7)66 60(sin x cos x)dx 8)2 304sin xdx1 cosx 9)4201 sin2xdxcos x 10)240cos 2xdx 11)261 sin2x cos2xdxsin x cosx 12)1x01dxe 1 .13) (cos x sin x)dx404 4 14) 401 2sin 2cos 2dxxx15) 20 2cos 3 1sin 3dxxx16)20 5 2sincosdxxx17) 022 2 34dxx x18) 112 x 2x 5dxBi 2:1)323x 1dx 2)421x 3x 2dx 3)53( x 2 x 2 )dx 4)222121x 2dxx 5)3x0 2 4dx 6)01 cos2xdx 7)201 sin xdx 8) x xdx202Bi 3:1) Tm cc hng s A,B hm s f(x) Asinx B tha mn ng thi cc iu kin' f (1) 2 v20 f(x)dx 42) Tm cc gi tr ca hng s a c ng thc :22 30[a (4 4a)x 4x ]dx 12 4. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt4II. PHNG PHP I BIN S :1) DNG 1: Tnh I =b'af[u(x)].u (x)dx bng cch t t = u(x)Cng thc i bin s dng 1: ( )( )( ) . '( ) ( )u bu abaf u x u x dx f t dt (1)Cch thc hin:Bc 1: t t u(x) dt u (x)dx ' Bc 2: i cn :( )( )t u at u bx ax bBc 3: Chuyn tch phn cho sang tch phn theo bin t ta c ( )( )( ) . '( ) ( )u bu abaI f u x u x dx f t dt (tip tc tnh tch phn mi)CH : +, Khi gp dng f(x) c cha (1, ln x)xth t t = lnx (v d 7, 9).+, Khi f(x) c cha n u(x) th thng t t = u(x).( v d 4,7, 5, 10 ...)+, Khi f(x) c mu s th thng t t = mu.Nhn chung l ta phi nm vng cng thc (1) v vn dng hp l.V d: Tnh cc tch phn sau:1)23 20cos xsin xdx ; 2)250cos xdx ; 3)420sin 4xdx1 cos x ; 4)13 20 x 1 x dx .5)22 30sin2x(1 sin x) dx ; 6)4401dxcos x ; 7)e11 ln xdxx ; 8)401dxcosx .9)e 211 ln xdxx ; 10)15 3 60 x (1 x ) dx ; 11)620cosxdx6 5sin x sin x ; 12).3 40tg xdx cos2x13)40cos sin3 sin2x xdxx ; 14) 202 2 cos 4sinsin 2dxx xx; 15) ln 5ln 3 2 3 x x e edx.16) 202 (2 sin )sin 2dxxx; 17) 34sin 2ln( )dxxtgx; 18) 408 (1 )tg x dx ; 19) 2 41 sin 2sin cosdxxx x.20) 2 0 1 3cossin 2 sindxxx x; 21) 20 1 cossin 2 cosdxxx x; 22) 20sin ( cos ) cose x xdx x ; 5. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt523) 21 1 1dxxx; 24) e dxxx x11 3ln ln; 25) 4 021 sin 21 2sindxxx.26)82311dxx x ; 27)7 33 201xdx x ; 28)35 20 x 1 x dx ; 29)ln2x01dxe 2 .30)733013 1xdxx ; 31)22 30 x x 1dx ; 32) 2 352 x x 4dx.2) DNG 2: Tnh I =baf(x)dx bng cch t x = (t)Cng thc i bin s dng 2: I f x dx f t t dtba( ) ( ) '( )Cch thc hin:Bc 1: t x (t) dx (t)dt ' Bc 2: i cn :ttx ax bBc 3: Chuyn tch phn cho sang tch phn theo bin t ta c I f x dx f t t dtba( ) ( ) '( ) (tip tc tnh tch phn mi)Ch :* Nu f(x) c cha:+, 2 2 n (a - x ) th t x = a . sin t vi t ;2 2- p p , hoc x = a .cos t vi t [0;p].+, 2 2 n (a + x ) th t x = a . tan t vi t ;2 2- p p , hoc x = a .cot t vi t (0;p).+, ( )2 2 n x - a th taxsin t= hocaxcos t= .V d: Tnh cc tch phn sau:1)120 1 x dx 2)1201dx1 x 3)1201dx4 x 4)1201dxx x 1 5)14 20xdxx x 1 6)2011 cos sindxx x 7)22 220xdx1 x 8)22 21 x 4 x dx 6. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt69)23221dxx x 1 10)3 2219 3xdxx 11)1501(1 )xdxx 12)222311dxx x 13)20cos7 cos2xdxx 14)1 46011xdxx 15)20cos1 cosxdxx 16) 012 x 2x 2dx17) 101 1 3xdx18) 2 1 51dxxx x.III. TNH TCH PHN BNG PHNG PHP VI PHN:* Kin thc:Cho hm s y = f(x) xc nh trn tp D vi phn ca hm s k hiu:dy = f '(x).dx hay d(f(x)) = f '(x).dx.* tnh c nhanh cc em cn nh nhng cng thc sau:+,d(a.x b)d(a.x b) a.dx dx (a 0)a++ = = .+,xx xxd(ae b)d(ae b) ae .dx dxa.e++ = = .+,d(sinx)d(sin x) cos x.dx dxcos x= = ;d(cos x)d(cos x) sin x.dx dxsin x= - =-.+,dxd(ln x) .x=dx 1 d(a.x b) 1ln(a.x b)a.x b a a.x b a+= = ++ +.+, 2 22 2x.dxd( x a )x a+ =+.V d 1: Tnh cc tch phn sau:1)10dx2007.x + 2008 ; 2)420sin x. cos xdx;p 3)e x2x1e .dx4 - 3e ; 4)46cot x.dxpp .V du 2: Tnh cc tch phn sau:1)1 23021x x ; 2)1 230( )2x x dx; 3)1 23021x x dx ; 4)210x xe dx ; 5)3121x x e dx .6)12 ln e xx dx ; 7)21 lneedxx x ; 8)330sincosxx dx ; 9)3cos0sin x x e dx ; 10)1x0dx2e + 3 .VI. TNH TCH PHN BNG PHNG PHP TCH PHN TNG PHN:Cng thc tch phn tng phn: 7. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt7 bababa u(x).v'(x)dx u(x).v(x) v(x).u'(x)dxHay: bababa udv u.v vduCch thc hin:Bc 1: t( )'( )'( )( )v v xdu u x dxdv v x dxu u xBc 2: Thay vo cng thc tch phn tng tng phn : bababa udv u.v vduBc 3: Tnh ba u.v v bavduCh :Gi s cn tnh tch phnba f(x)g(x)dx ta thc hint u = f(x), dv = g(x)dx (hoc ngc li) sao cho d tm nguyn hmv(x) v vi phn/ du = u (x)dx khng qu phc tp. Hn na, tch phnba vdu phi tnh c.c bit:i/ Nu gpb b baxa a a P(x) sin axdx, P(x) cos axdx, e .P(x)dx vi P(x) l a thc th t u = P(x) .ii/ Nu gpba P(x) ln xdx th t u = ln x .iii/ Nu gpbxae . sin axdx a ,bxae . cos axdx a th ta tnh hai ln tng phn bng cch t x u ea = ..V d: Tnh cc tch phn sau:1)251ln xdx x 2)220x cos xdx 3)1x0e sin xdx 4)20sin xdx 5)e21x ln xdx 6)320x sin xdxcos x 8. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt87)20xsin x cos xdx 8)420x(2cos x 1)dx 9)221ln(1 x)dxx10)12 2x0 (x 1) e dx 11)e21(x ln x) dx 12)20cosx. ln(1 cosx)dx 13) 21ln( 1)eexdxx 14)120xtg xdx 15) 102 (x 2)e dx x16) 102 x ln(1 x )dx 17) edxxx1ln18) 203 ( cos ) sinx x xdx19) 20(2x 7) ln(x 1)dx 20) 322 ln(x x)dxC. MT S BI TON TCH PHN QUAN TRNG V NG DNGBi 1: 1) CMR nu f(x) l v lin tc trn [-a;a] (a>0) th :aaf(x)dx 0 2) CMR nu f(x) chn v lin tc trn [-a;a] (a>0) th :a aa 0f(x)dx 2 f(x)dx .V d: Tnh tch phnI=222cos x. ln(x x 1)dxp- p + +Bi 2: 1) CMR nu f(x) l mt hm s lin tc trn an [-a; a] vi a > 0 th:a aa 0f(x).dx (f(x) f( x)).dx- = + - .V d: Tnh tch phnCho f (x) l hm s lin tc trn R tho mn f (x) + f (- x) = 2 - 2.cos2x .Tnh tch phn3232I f(x).dxp- p= Bi 3: Nu hm s f(x) lin tc trn on [ 0; a] vi a > 0, tha a0 0 f(x)dx = f(a - x).dx .Bi 4: Nu hm s f(x) lin tc trn on [a; b] v tho mn f(x) = f( a +b - x) thb ba aa bx.f(x)dx . f(x).dx2+ = 9. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt9H qu: a)2 20 0f(sin x)dx f(cosx)dx b)0 0xf(sin x)dx f(sin x)dx2 .V du: Tnh tch phna) 20I x. sin x. cos x.dxp= ;22201b)J ( tan (sin x)).dxcos (cos x)p= - .Bi 5: Nu f (x) l hm s lin tc, tun hon c chu k T th :Ta T T 2a 0 T2f(x)dx f(x)dx f(x)dx, a R+- = = " .V d: Tnh cc tch phna)220I ln(sin x 1 sin x)dx;p= + + b)200820070J sin x.dxp= .Bi 6:CMR nu f(x) lin tc v chn trn R th+0( )( ) vi R va a > 01 xf xdx f x dxa ; a 1V d : Tnh cc tch phn sau:a)1 412 1 xxdx b)1 2111 2xxdx c)2 sin3 1 xxdx P DNG: Tnh cc tch phn sau:1)2 n+n n0cos xdx vi n Zcos x sin x ; 2)2 44 40cos xdxcos x sin x ; 3) .2 66 60sin xdxsin x cos x 4)50xsin xdx ; 5)2224 sinx cosxdxx ; 6)1 421sin1x xdxx ; 7) 20xsin xdx4 cos x 8)4 30x cos x sin xdx ; 9)1 2008x1xdx2007 1-+ ; 10)420080log (1 tan x).dxp + .11)4 6 6x4sin x cos x6 1p- p++ ; 12)420x. sin xdx2 cos xp+ . 10. GV: TRN PHONG Khai ging lp mi hng nm vo ngy 30/6 77 N Trang Gh - bmtTel: 0927.244.963 www.facebook.com/phongmath.bmt10D. PHN LOI MT S DNG TCH PHNI.TCH PHN LNG GIC1. Dng bc l vi hm sin.Phng php chung: t t = cosx khi dt = - sinx.dx, sau a tch phn ban u v tchphn theo bin t.Ch :2 2 22n 1 2 n 2 nsin x 1 cos x 1 t .(sin x) (sin x) . sin x (1 t ) . sin x += - = -= = -V d 1 (bc sin l). Tnh tch phn22 30I cos x sin xdxp= .Giit t = cos x dt = - sin xdxx 0 t 1, x t 02p= = = =2 02 2 2 20 1I cos x(1 cos x) sin xdx t (1 t )dtp = - = - -1 1 3 52 40 0t t 2(t t )dt3 5 15 = -