CHUONG III: NGUYEN HAM, TICH PHAN VA UNG DUNG A. KIEN THUC CO BAN 1. NGUYEN HAM - Nguyen ham: Cho K la mot khoang (a;b), nua khoang (a;b], [a,b) hay d

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f(), V e K Neu F() la mot nguyen ham cua f() thi hp cac nguyen ham

ham: = C - = In C Voi a * - ta co:..a c a.

voi k la hang so f = In u C u.a fu".u' = C a cosu.u'. = sinu C f "sinu.

u' = e u c i In a c r a u a u.u'. = c (a>0,a*l) In a - Tinh chat co

[g() kf()=kjf( ) voi moi so k - Phuong phap doi bien so: Dang : Neu

1 CHUONG III: NGUYEN HAM, TICH PHAN VA UNG DUNG A. KIEN THUC CO BAN. NGUYEN HAM - Nguyen ham: Cho K la mot khoang (a;b), nua khoang (a;b], [a,b) hay doan [a;b]. Ham so F() goi la mot nguyen ham ciia ham so f() tren K niu: F'() = f(), V e K Neu F() la mot nguyen ham cua f() thi hp cac nguyen ham ciia f() la: f() = F() C, C la hang so bat ki - Bang cac nguyen ham: = C - = In C Voi a * - ta co:..a c a. C a cos = sin C sin = -cos C = tan C cos cot C sm" e = e* c k = k C voi k la hang so f = In u C u.a fu".u' = C a cosu.u'. = sinu C f "sinu.u'. =-cosu C f u' 5 = tanu C cos u f ^5 = -cotu C sin u [e u.u' = e u c i In a c r a u a u.u'. = c (a>0,a*l) In a - Tinh chat co ban: Neu f va g la hai ham so lien tuc tren K thi: (f () g()) = f() [g() kf()=kjf( ) voi moi so k - Phuong phap doi bien so: Dang : Neu = u(t) co dao ham lien tuc tren K thi: f()= f(u(t)).u'(t).dt Dang : Neu t = v() co dao ham lien tuc tren K va co f() = g(t)dt thi: f() - g(t)dt

- Phucmg phap nguyen ham tung phan: Neu u(), v() co dao ham lien tuc tren K thi udv = uv - vdu B.

g() (f () - g()) = f () - g() j"kf () =k f () voi moi so k Chii y: - Phoi hop diing bang cong thiic

viet mu m phan so va m = a n - Voi ham huu ti, neu bac ciia tu Ion hon hoac bang bac ciia mau thi

Neu bac cua tu be hem bac ciia mau thi bien doi ve tong, hieu ciia cac don thuc va phan thiic don

$nguyen ham ciia f(): a)f()= l-cotf ^l ; f() = \ 4) sin b) F() = ln - ; f() = ln c) F() = e sin3 ;$

2 - Phucmg phap nguyen ham tung phan: Neu u(), v() co dao ham lien tuc tren K thi udv = uv - vdu B. PHAN DANG TOAN DANG : BINH NGHlA VATTNH CHAT - Be chung minh F() la mot nguyen ham ciia f() ta chiing minh F'() = f() voi moi thuoc D. - Sir dung tinh chat chit co ban: Neu f va g la hai ham so lien tuc tren K thi: (f () g()) = f () g() (f () - g()) = f () - g() j"kf () =k f () voi moi so k Chii y: - Phoi hop diing bang cong thiic voi cac bien doi chia tach, them bat, khai trien tich so, hang dang thiic, nhan chia lucmg lien hap, viet mu m phan so va m = a n - Voi ham huu ti, neu bac ciia tu Ion hon hoac bang bac ciia mau thi phai chia tach phan da thuc, con lai ham huu ti vdi bac tir be ban mau. Neu bac cua tu be hem bac ciia mau thi bien doi ve tong, hieu ciia cac don thuc va phan thiic don gian bang each dong nhat he so hoac them bat, chia tach,... it-i L ~) "-a L (-a)( a) a -a a, A B hay r=- - = -a" (-a)( a) -a a Vi du : Chung minh F() la mot nguyen ham ciia f(): a)f()= l-cotf ^l ; f() = \ 4) sin b) F() = ln - ; f() = ln c) F() = e sin3 ; f() = 3e s,n3. cos3 a)f'()= }, =, =-±- =f()=>dp. n, TC i sm sm 4) -cos {. b) F '() = ln. - = ln = f() => dpcm. c) F '() = e sm3 3cos3 = f() => dpcm. 47

$c)f'() = V _ dpcm. V V cos => dpcm.. sm tan- cos sin dpcm. f f cos' - - tan 4 u 4j r TC N \.$ f TO cos sin sin rc cos v 4, U 4) Vi du 3: Chung minh rang cac ham so sau deu la mot nguyen ham ciia ciing mot ham so, chi ra ham so do: 6 l 0 a) F() = va G() =

f TO cos sin sin rc cos v 4, U 4) Vi du 3: Chung minh rang cac ham so sau deu la mot nguyen ham ciia ciing mot ham so, chi ra ham so do: 6 l 0 a) F() = va G() =

3 Vi du : Chung minh F() la nguyen ham cua f(): a) F() = ln( vl ) C b) F() = In c) F() = In tan C r tan X TC v 4 ; m ; f() c ; f() = vt~^ i sin cos a) F '() b) F '() c)f'() = V _ dpcm. V V cos => dpcm.. sm tan- cos sin dpcm. f f cos' - - tan 4 u 4j r TC N \. f TO cos sin sin rc cos v 4, U 4) Vi du 3: Chung minh rang cac ham so sau deu la mot nguyen ham ciia ciing mot ham so, chi ra ham so do: 6 l 0 a) F() = va G() = -3-3 b) F() va G() = 0 cot. sm a) Vi F() = X^ X^ 6 = X ^ 3 = G() 3, nen F() va G() deu la mot nguyen ham ciia f() (-3) sincos cos b) ViG'() sin sin 3 - cos G'() = cot nen F(), G() deu la mot nguyen ham sin" sm" cos ciia f() sin 3 48

Vi du 4: Tim nguyen ham ciia cac ham so sau: a) f() = 3 - c)f() = (- l)( 4 3) b) f() = 3-5 7 d)f() = (-9) 4 a) f() = jf3 -] = 3 C b) f() = "( 3-5

$a b Khi a = - thib = (a b-ln a b ) C a (a b) a (a b) Khi a 5t-,- thi B = a a a Vi du 6: Tim nguyen ham ciia cac ham so sau: C a)f() = 3 -\ - X" X$

4 Vi du 4: Tim nguyen ham ciia cac ham so sau: a) f() = 3 - c)f() = (- l)( 4 3) b) f() = d)f() = (-9) 4 a) f() = jf3 -] = 3 C b) f() = "( 3-5 7) C c) ( - l)( 4 3) = ( ) (-9) 5 d) (-9) 4 C. o 3 _ C 6 5 Vi du 5: Tinh: a) A = f( - 3) b)b= f(a bf.,a^0. a) A = ( - 3 3)( - 3) = }(-3) 3 (-3) n = (X " 3)3 (X_3)Z 3 C b) Ta co B = f (a b) a - (a b) a Khi a = - thi B = In la b C v. a b Khi a = - thib = (a b-ln a b ) C a (a b) a (a b) Khi a 5t-,- thi B = a a a Vi du 6: Tim nguyen ham ciia cac ham so sau: C a)f() = 3 -\ - X" X 4 3 c) f() b) f() (-) d)f() = (l ). 49

5 5 3 5 ) j(3 -^ -] 3 - -=- = ln C. ) b) f L^=-^ c (-) - d) (l - " 3 ) = ( - ~ ) = - C. Vi du 7; Tim nguyen ham cua cac ham so: a) f() = 8 - A b) f() = V X N c,, c) f() = d) f() (-l) - r r 8 3 a) f() = j(8-. 4 ) = C 7-, ff,, b) flz^= ff 3 i -7 = 3 ln-7 C 9_V Y l, O -7 7' c)tacof() = -l nen ff()= ~- ^- C (-l) -l 3 r d) Ta co f() = nen f() = 3n - C - ' Vi du 8: Tinh:,, r } f ^ L _ d b) U ; ~ (-)( 3) l_ : a) Dat - ^ = L nen = A( 3) B( - ) (-)(3) - 3 Do do = (A B) (3A - B), dong nhat he s6 thi: A B =,3A-B = nena= -,B =

$( ) X X dong nhat thi co: - d) ff()cl= f-- i r l = -ln l-- hi l C = In \ X X lj X Cach khac: Bien doi = - -. 4 - - - = "' 3 = - (-l)( l) X -X X - - X Dat?$

6 l Do do <v- -a (-)( 3) 3 5(-) 5( 3) :-ln - -ln 3 C 5 5 b) j-^dx: = In I I In I I C=ln l- - A B C ^ c) Dat L,. ( ) X X dong nhat thi co: - d) ff()cl= f-- i r l = -ln l-- hi l C = In \ X X lj X Cach khac: Bien doi = = "' 3 = - (-l)( l) X -X X - - X Dat? = - nen - = (A B C) (B - C) (-l)(l) -l dong nhat he so thi duoc A =, B =. C =. do do: ff()=l X -l Vi du 9: Tinh - n - In I z - I C. a) [(Vv) b)f V V f i a) (v v) = V b) iw^v V V j n 3 = 3 3 C 3' 3 N X = V --= C d) j7(\/ - V ) i A = V--V Y C ( 5 3 ^ d) v(v-v" D = X 6 -X 4 = 6 4 C 7 3

$5 -BDHSG DSGT/ Vi du 0: Tinh ck ck 0 I = f r b)= f, a * 0, b * c V3-V-4 VabVac :)E= jv 4 " 4 d) F = { V I 3 3 a) = ^ V 3v^4)= j^(3) (-4) = ($ 3 r - = f ( )^_( ) 3 7 = ( )s -3( )f C Vi du : Tim nguyen ham ciia cac ham so: a) f() = cos c) f() = tan b) f() = sin d) f() = cot f cos =

3 r - = f ( )^_( ) 3 7 = ( )s -3( )f C Vi du : Tim nguyen ham ciia cac ham so: a) f() = cos c) f() = tan b) f() = sin d) f() = cot f cos =

7 5 -BDHSG DSGT/ Vi du 0: Tinh ck ck 0 I = f r b)= f, a * 0, b * c V3-V-4 VabVac :)E= jv 4 " 4 d) F = { V I 3 3 a) = ^ V 3v^4)= j^(3) (-4) = ( 3) (-4) C b) = \/ab-\/acj (a b) -^(a c) C a(b-c) \ _ 3 c) E = jv( ) = j^ \ C. 3 r - = f ( )^_( ) 3 7 = ( )s -3( )f C Vi du : Tim nguyen ham ciia cac ham so: a) f() = cos c) f() = tan b) f() = sin d) f() = cot f cos = sin C -cos sin b) jsin = = - - ^ ^ C 4 c) ftan = [( = tan - C Vcos" ) d) [cot = [( - ) = - cot - C > sin" Vi du : Tinh: a) I - sin3cos b)= cos5cos7 d)k= c) j"sin H = 4 jcos 3 a) I = (sin sin5) = -i(cos cos5) C

$b) - i (cosl cos) = ^-(-^sinl sin) C c) H = f(cos3 3cos) = sin3 sin C 4 4 d) sin 4 = ( - cos) = ( - cos cos ) 4 4 3 = ( cos cos4) 4 3 nen K = ( - sin sin4) C 4 8 Vi du 3: Tinh: \ f i,,$ rl-cos rsin" / o) = = = (tan-) C cos cos -"Icos X ' X j sin cos cos c) tan cot z = h = : = nen cos sin cos sin sin sin4sin3.

rl-cos rsin" / o) = = = (tan-) C cos cos -"Icos X ' X j sin cos cos c) tan cot z = h = : = nen cos sin cos sin sin sin4sin3.

8 b) - i (cosl cos) = ^-(-^sinl sin) C c) H = f(cos3 3cos) = sin3 sin C 4 4 d) sin 4 = ( - cos) = ( - cos cos ) = ( cos cos4) 4 3 nen K = ( - sin sin4) C 4 8 Vi du 3: Tinh: \ f i,, rl-cos a) K-s = b). r sin sm^sincs cos, ' r cos sin4sin3, N f sin - cos c) -, IT 77T d ) sin tan cct r 0 f. 9 ~ = ff ^ = tan-cot C ' sin" Gin" vfi cos" V cos" sm. rl-cos rsin" / o) = = = (tan-) C cos cos -"Icos X ' X j sin cos cos c) tan cot z = h = : = nen cos sin cos sin sin sin4sin3... = sin 4 sm 3 sm z = sm 4(cos o - cos ) tan cot = (sin4cos5 - sin4cos) = - (sin9 - sin - sin5 - sin3). Vay f 5^8^ ^ = _ jl (_ i_ COs9 cos cos5 cos3) C tan cot ,v rsin -cos l-, / V _, i i _ d) = = In cot C sin v sin j Vi du 4: Tinh: ) L*-\) b) l0 e' c) ( -3 ) d) 5 " X ) 53

$6 9 ) = - C In 4 In 6 In 9 V ; f5> > - -3" = v3y 5 T In 5 ln3 3 Vi du 5: Tinh: \ T _ f COS X f a ) _ : b)e= cos.cos3 cos sin a)xet= [ thi: I = f = C, cos sin T T _ fcos - sin. j.$

9 54 a) j e*- j = [(e e-- ) = e -e' C b) fl0 =fl00 = ^ C=-^ C ' In 00 nl0 c) f( -3 ) = ( ( ) = - C In 4 In 6 In 9 V ; f5> > - -3" = v3y 5 T In 5 ln3 3 Vi du 5: Tinh: \ T _ f COS X f a ) _ : b)e= cos.cos3 cos sin a)xet= [ thi: I = f = C, cos sin T T _ fcos - sin. j. i-- : = In cos sm C 9 cos sin Suy ra I = ( In cos sin ) C. b)xetf= sin.cos3 thi:e F = [cos3 =^-sin3 C, 3 X C. E - F = jcos. cos 3 = - (cos 5 cos ) = sin 5 sin C 9 0 Suy rae = -^sin5 -sin3 -sin C Vi du 6: Tinh: a) (l tan) e b) [V a) (l tan) e = f(l tan tan)e = (tan.e ) =tan.e C mm nphj ^g^r

b) Xet f() = (a 4 b 3 c m)e thi f'() = (4a 3 3b c d)e (a 4 b 3 c m)e = [a 4 (4a b) 3 (3b c) (c d) d m]e Ta co f'() = 4 e <=> a =, 4a b = 0,

DANG : PHUONG PHAP BIEN B6l BIEN S6 Dang : Neu = u(t) co dao ham lien tuc tren K thi: jf()= jf(u(t)).u'(t).

Md rong cong thuc thanh u kem theo du = u'., luu y dau cong trir va he so nhan chia.

$tant " -t- a" Dang \la - thi dat = asint hay = acost Dang V a thi dat = atant.$

10 b) Xet f() = (a 4 b 3 c m)e thi f'() = (4a 3 3b c d)e (a 4 b 3 c m)e = [a 4 (4a b) 3 (3b c) (c d) d m]e Ta co f'() = 4 e <=> a =, 4a b = 0, 3b c = 0, c d = 0, d m = 0 oa = l,b = -4, c =, d = -4, m = 4. Vay V = ( )e C. DANG : PHUONG PHAP BIEN B6l BIEN S6 Dang : Neu = u(t) co dao ham lien tuc tren K thi: jf()= jf(u(t)).u'(t).dt Dang : Neu t = v() co dao ham lien tuc tren K va co f() = g(t)dt thi: f() = g(t)dt Chii y: - Su dung cac cong thuc mo rong k voi k * 0. Md rong cong thuc thanh u kem theo du = u'., luu y dau cong trir va he so nhan chia. - Neu f() = F() C thi f(u)du = F(u) C - Chon dat bien thich hop, bien u ma bieu thuc co san u' - Doi bien dai so: u = g(), = h(t), t =,... Dang, ' V k thi dat t = %/ k - Doi bien luong giac: Dang 9 9 thi dat = a.tant " -t- a" Dang \la - thi dat = asint hay = acost Dang V a thi dat = atant. Dang V -a a thi dat = sin t, Bien doi theo goc phu t = tan Bac le voi sin thi dat t = cos, bac le vdi cos thi dat t = sin Dac biet d6i bien t = -, t =TC -, t = -. - Nguyen ham lien ket, de tinh I thi dat them ma viec tinh I va I -, I k va I - m thuan loi hon, tu do suy ra I 55

11 Vidu : Tinh: a) f( l) 9 b) j(3 - ) 5 c) j" 3 (l 4 ) 3 d) f 8 a) Doi bien u = thi du = => = du f( ) 9 = - fu 9 du-..u 0 C= ( l) 0 C 0 0 Cach khac: Viet gop j( ) 9 = - ( l) 9 d( ) = L( l) 0 C b) Dat u = 3- => = 3- u=> = -du (3-5) 5 = - (3 - u).u 5.du = (u 6-3u 5 )du = iu 7-3 -' u C= (3-) 6 C 7 6 c) f( 3 (l 4 ) 3 =i (l 4 ) 3.d(l 4 ) = ^( 4 ) 4 C 3 d) Dat u = ^> du = => = 6du = 6 fu f 57 du = u 5 8 C 8 9 9,8 Vi du : Tinh a) I l - - X X d) - d -a b). d ) _ 3 5 a) Doi bien: u = thi du = ( l). rdu f-^- = f = lnu C = ln( l) C X" u 58 C :) f - (l a ) X X 56

12 b) Dat u = thi du = => = -du rdu -.- C '( ) "" -V "u ( ) C, f _ r 7 r d( 8 ) _. 8 j '(l 8 )" 8 (l 8 )"8 8 (l 8 )"8 "l 8,, - a d ) f - r ^ r ^ f l - = In ) a a c -a" a >,-a a - ^ ff -3 X A -- X X d - Vl " = - In r-: C In - z X C Vi du 3: Tinh: a) [sin 4 cos c) COi -- a) Dat t = sin thi dt = cos b) j"cos( ) d) f^sin cos " [sin 4 cos = [t 4 dt = t 5 C = Sm X C. 5 5 b) Dat t = thi dt = f cos( ) = ("cos t.dt = sin t C = sin( ) C c) Dat t = - - thi dt = r- => -4- = -dt j-^cos^" " - j = - cost.dt = - sint C = -sin^ - j C d)datt = sin-^>dt A rr.cos dx 57

$Do do: f 5-sin cos = - [t.dt = -^t C = --sin C Vi du 4: Tinh: f 3 >\ a) A = sin X b)b k i cos sin - c) M = jsin 3 cos 4 XT rsin 3.$

13 Do do: f 5-sin cos = - [t.dt = -^t C = --sin C Vi du 4: Tinh: f 3 >\ a) A = sin X b)b k i cos sin - c) M = jsin 3 cos 4 XT rsin 3. d)n= : COS X I 3 i a) Dat u = => du = => = du 3 z r r - A = sinudu = cosu C L = cosf ) C b) Doi bien: u = sin thi du = cos rdu I,,,., B = -f- - In u C = In sm C. > u I I I c) Dat u = cos thi du = -sin M= (l -cos )cos 4. sin = -f(l-u )u 4.du= f(u 6 -u 4 )du= u 7 --u 5 C= -cos 7 -- cos 5 C d) Dat u = cos thi du = -sin ( cos" )sin K - f du cos 4 -l u 4 -u- 3 -i C 3 3cos cos C d u = Vi du 5: Tinh: sin cm 3 Y sin cos X sin sincos (a > 3/ ~ 5t b ) vi cos X va sm" b cos a) Dat u = cos thi du = -sin r = rsm = _r du sin v sin 4 (l-u ) u 58 -BDHSG DSGT/

$Ta co: _ [(u l)-(u-l)f (-u ) (u-l) (u l) 4' (u-l) (u l) f (u ) - (u-) u -l, cos nen : [ \ = In tan sin 3 9 sin C b) Ta co: sin sm o A fl sin, COSX o X X nen = tan ZCOS COS cos c)$ Va sin b cos a b a -b c Vi du 6: Tinh: / " r~ 9 ^ 5 -va sin b cos C a -b a) jcot b) j"tan 3 cot :) fcot 5 d) f sin 9 a) Dat u = sin thi du = cos fcot = f^^ = [ du = ln u C = ln

Va sin b cos a b a -b c Vi du 6: Tinh: / " r~ 9 ^ 5 -va sin b cos C a -b a) jcot b) j"tan 3 cot :) fcot 5 d) f sin 9 a) Dat u = sin thi du = cos fcot = f^^ = [ du = ln u C = ln

14 Ta co: _ [(u l)-(u-l)f (-u ) (u-l) (u l) 4' (u-l) (u l) f (u ) - (u-) u -l, cos nen : [ \ = In tan sin 3 9 sin C b) Ta co: sin sm o A fl sin, COSX o X X nen = tan ZCOS COS cos c) Dat u = cos thi du = -sin { ^ ^ d = - -^L = - fu^du = -3u3 C = - S 3 /^ ' 3/ VCOS X -b ) sin cos d) Dat u = Va sin b cos => du = (a 4 - i a sin b cos sm cos - = f \ f d u =, 9 u Va sin b cos a b a -b c Vi du 6: Tinh: / " r~ 9 ^ 5 -va sin b cos C a -b a) jcot b) j"tan 3 cot :) fcot 5 d) f sin 9 a) Dat u = sin thi du = cos fcot = f^^ = [ du = ln u C = ln sin C sm u b) tan 3 = [tan(l tan - l) f rsin Q. - tan(tan)- = tan In cos C cos. r E, rcos 5, r(l -sin ). c) cot 5 = r = U(sin) } > sin sin 0 r, d(sin) = - sm sm sm 4sin sin~ ln sin3

15 cos.sin d(sin 9 ) d) M=i-d*=- = ^ Ik sm ' s sin 9 (l sin 9 ) 9 sin 9 (l sin 9 ) rd(sin 9 ) rd(sin 9 ) fd(sm _ ) 9 F sm 9 9 > sin 9 Vidu7: Tinh: a)a= V7-3 -In 9 sin 9 sin C b)b= j" ( - 3)V~ r 3 c)q= f-^r - V a) D6i bien: t = 7-3 thi dt = -6 d)p= j V 4 A = - - fyt.dt = --tc = --(7-3 )C 4 ' 9 3 b) Dat t = V-3 => = t 3 => = t.dt B = (t 3)t dt = (t 4 3t )dt = -t 5 t 3 C = -(-3)l(-l) C. 5 5 c) Dat t = - V" => = ( - t) => = -( - t)dt Q= jl-_idt = ^-ijdt = (t-ln t ) C = -(V ln - V ) C i 3 d) P= f_^ = = - f( 4) 3d( 4) = -( 4) 3 C,3 3/_ / 4. A 9. 4 Vi du 8: Tinh: b) f ± «k o f /(l V) X ; )f, V 9 ( l) (l e ) a) Dat u = V f V(l V) du V" u" u C = V c 60

ln V 9 C r( >( l) _ rf r dt = ln '(l t(l e ) ~~ l t- t t- b) j*e sin cos d) f 'e -e a) Doi bien: u = - thi du = - j"e " = - jv'du b) jv m

16 b) Dat t = Vl rvt = t - => = t.dt dt [dt ~ ) d t = t l n I* _ l "! C VITl : ln^^^c. Vl c) Dat t = V 9 =>dt = rdt r a _ rdt KL 'V -Q ~ t 9 d) Dat t = e thi dt = ( l)e Vi du 9: Tinh: a) je~ r(ln)~, ^-, dt V 9j V 9 t ln t C = ln V 9 C r( >( l) _ rf r dt = ln '(l t(l e ) ~~ l t- t t- b) j*e sin cos d) f 'e -e a) Doi bien: u = - thi du = - j"e " = - jv'du b) jv m cos = e sm d(sin) = e sm C t V c = -- e - c C = In e' e C C) jcl^ol d = ( l n ) d ( l n ) = ln3 c d) Dat t = e thi dt = e => = -dt t e -e _ t(t-t _ ) t - ^t-l t. Idt = i(ln t-l -ln t l ) C =-ln e" - e C

Vi du 0: Tim ham so y = f() biet: a)dy = (3 - l) 3 vaf(l) = 7 3 b) y' = sin. cos va f(n) = 79. a) Dat u = 3 - thi du = 6 nen n = du.

b) Dat t = sin thi dt = cos f()= j"y'= jsin 7 (l-sin ) cos = jv(l-t )dt sin 8 sin 0 = f(t 7 -t 9 )dt=-l-^ C = ^ - ^ ^ C 8 0 8 0 8 0 RTY Y sin V Vi f(tt) = 79 nen C = 79. Vay f() = - 79.

$tiep,... - Cac dang ham hon hop e\sin, e cos, P().e,.ln,.ln, P().ln,... deu dung phuong phap tirng phan, co the dat u, dv lien tiep nhidu lan.$

17 Vi du 0: Tim ham so y = f() biet: a)dy = (3 - l) 3 vaf(l) = 7 3 b) y' = sin. cos va f(n) = 79. a) Dat u = 3 - thi du = 6 nen n = du. (3 -l) 4 f()= dy = u 3 du = -y C = v d X ~ i C Vi f(t) = 8 C = 3 nen C = -5. Vay f() = (3 ~~ )4-5. b) Dat t = sin thi dt = cos f()= j"y'= jsin 7 (l-sin ) cos = jv(l-t )dt sin 8 sin 0 = f(t 7 -t 9 )dt=-l-^ C = ^ - ^ ^ C RTY Y sin V Vi f(tt) = 79 nen C = 79. Vay f() = Vi du : Dat I n = jtan", n e Tinh L, theo I n _, n > 3. Vdin>3:I n = [tan n_ tan = "tan n " (tan = ("tan - (tan) - I n _ = - tan " - I n _. n-l DANG 3: NGUYEN HAM TUNG PHAN - l) Neu u(), v() co dao ham lien tuc tren K thi udv = uv - vdu Chu y: - Chon dat u va dv de dua ve nguyen ham gan cong thuc hon hoac dang vong tron, tung phan lien tiep,... - Cac dang ham hon hop e\sin, e cos, P().e,.ln,.ln, P().ln,... deu dung phuong phap tirng phan, co the dat u, dv lien tiep nhidu lan. - Voi tich phan truy hoi I n theo I n -i hay I n _ thi sin,, cos" tach luy thira va dung phuong phap nguyen ham tirng phan con tan", cot" tach luy thua va dirng phuong phap nguyen ham doi bien so. Vi du : Tinh: a) fcos b) [ \ sin a) Dat u() =, v'() = cos. Khi do u'() =, v() = sin Ta co jcos = judv = uv - jvdu = sin - sin = sin cos C 6

dt = (-tcost costdt) = (-tcost sint) C = ( V cos V sin V ) C Vi du 3: Tinh: a) ln b) V.ln a) Dat u = ln, dv =. Khi do du =, v =. Ta co: ln = ln -.

18 b) I * = - (d(cot ) = -cot (cot sm rcos i i = -cot =-cot In sin C Vi du: Tinh: sin a) I = cos b) = sinv a) Dat u =, v' = cos. Khi do u' =, v = sin. I = ^ sin- "sin = ^ sin i d(cos) = sin cos - sin C. 4 b) Dat t = V => = t => = tdt = smt.t.dt = jt.sint.dt = (-tcost costdt) = (-tcost sint) C = ( V cos V sin V ) C Vi du 3: Tinh: a) ln b) V.ln a) Dat u = ln, dv =. Khi do du =, v =. Ta co: ln = ln -. = ln - = ln - C. i - b) Dat u = ln, v'= v ^>u'=.v= Taco: 3 fvln = ln - f = ln C Vi du 4: Tinh: a) f 3 ln() b) fln ^ a) Dat u = ln(), v' = 3 Khi do u' = -. v = 4 3 ln() = udv = uv - vdu 4 = - 4 ln() - - f 3 = i 4 ln() - C

19 b) Bat u= In. du =. Khi do du =. v = (l ) f In X = In f X l - / V,,,, n In - = In -ln C -V ) Vi du 5: Tinh: a) I = fe b) = j 3.e a) I = fe = e - jv = e - e C = e ( - ) C b) Dat u = 3, v' = e thi = 3.e = e 3-3.e Dat u =, v' = e thi f.e = e - je = e - Do do = e ( ) C. Vi du 6: Tinh: a) A = fe^^ b) B = jln( Vl ) Q a) Dat t = V3-9 => 3 = t 9 => = -tdt 3 A = jte'dt. Dat u = t, v' = e l thi te'dt = e' - e' C nena= (V3-9 e ^ - e 7 3^ j C b) Dat u = ln( Vl ), dv = khi du =- r =i= Vl v =. B = ln( Vl )- = ln( Vl )-Vl C /l z ln(sin) Vi du 7: Tinh: a) f ln ( sin ) b) f X sin cos. rirnsin; Tn(sin) r.., i) cos T, - ln(sin) d(tan) * rric v = tan.ln(sin) - [ = tan.ln(sin) - C. b) Dat u =, dv = sin = ( - cos) Khi do du =, v = ( - sin) -BDHSG DSGT/Z-

jsin = udv = uv - jvdu = (- sin) - jj - sin j l. = f - sin) - ( cos) C 4 - o o n sm cos L 4 4 8 Vi du 8: Tinh: a) I = jsin(ln) b) = e (cos sin) a) Dat u = ln thi = e u nen = e u du A = jsinu.

cos = e.sin-e.sin nen = j"e (cos sin) = e.sin C Vi du 9: Dat I n = [ sin n, n e N Tinh I n theo I n _ va n > 3. I n = Isin""^.sin = sin n_ d(cos) = -sin n_.cos (n - ) jsin n ".cos = -sin n_.

20 jsin = udv = uv - jvdu = (- sin) - jj - sin j l. = f - sin) - ( cos) C 4 - o o n sm cos L Vi du 8: Tinh: a) I = jsin(ln) b) = e (cos sin) a) Dat u = ln thi = e u nen = e u du A = jsinu.e u du = sinud(e u )= sinu.e" - jcosu.e u du = sinu.e" - cosu.d(e u ) = sinu.e - cosu.e u - sinu.emu Tu do suy ra A = (sin(ln) - cos(ln)) C b) Dat u = e", dv = cos. Khi do du = e ', v = sin je.cos = e.sin-e.sin nen = j"e (cos sin) = e.sin C Vi du 9: Dat I n = [ sin n, n e N Tinh I n theo I n _ va n > 3. I n = Isin""^.sin = sin n_ d(cos) = -sin n_.cos (n - ) jsin n ".cos = -sin n_.cos (n - ) sin n_ (l - sin ) = -sin n_.cos (n - )I n - - (n - l)i n Do do I n = - sin -.cos -I n _ n n Vi du 0: Dat I = f n e, n e N* Tinh I theo I n _i voi n >. Suy ra I 3. I n = "d(e) = n.e - n f^e' = n.e - nl _i Do do I 3 = 3 e - 3I, h = e -,, I, = e = e ( - ) C nen I 3 = e ( ) C.

NO CIIOONG THUONG NIEN NAM 2017 DAI H(.M DONG COn Vietcombank ran HANiOf-JMCP DAI THlTONG VIET NAM C0NG HOA XA HOl CHU NGHIA VIET NAM Doc lap - Tur do

NO CIIOONG THUONG NIEN NAM 2017 DAI H(.M DONG COn Vietcombank ran HANiOf-JMCP DAI THlTONG VIET NAM C0NG HOA XA HOl CHU NGHIA VIET NAM Doc lap - Tur do - Hanh phuc Dia< M3D1 16/12/201 ~.98 T*. Khai,HaN6i