Ôn tập nguyên hàm và tích phân

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1<2f'W;'g()5 2 Lê Xuân Manh  §T: 0373840709 a/ f(x) = H H E:HZ]H:^Z]Zc biết F( ^ π )= - 3 b/ f(x) = ^ H H x x x − − biết F((-3) = 10 HXH,'()*+#6'47'4489:; 12YVW&$%=49'&$%9! >$%?@=") !8 A  \K(OB([ K OF u C+ ∫ 5  \n(KZOpBZ[ n K OpF u x C+ ∫ W,'()*+#./+#:;:/( / E:Z :e xdx ∫  h Kh ^Ox dx+ ∫  ^ H E: tg x dx x ∫ B :Z E:e xdx ∫ W, / H ^ r dZ x dx ∫  H ` x x dx− ∫  H  K O dx x x+ ∫ B   x dx e + ∫ W, / h Z : E: ^ ^ x c dx ∫  ^ H ` x x dx− ∫  H   dx x x− ∫ B HY ^x dx x + ∫ W, /  /Zdx ∫  E Zdx ∫  ^ K/  /ZOdx+ ∫ B ^ KE Z]E ZOdx ∫ g/ H  s r dx x x− + ∫ h/ H ^ H s r x dx x x + + + ∫ HXX^,'()*+#6'47'44='4! 12YVW3"8'!95  (KZO<tKZOBZ[ K O K O K O tK Ou x v x v x u x du− ∫ ∫ W,'()*+#./+#:;:/( /\KZO[Z H :HZ\KZO[Z H E:Z \KZO[Z H e Z B\KZO[Z ^ YKHZO W, a/ :Zx dx ∫ b/ Z K Ox e dx+ ∫ c/ KH O:x xdx+ ∫ (x) d/ ` : E:x xdx ∫ e/ h Kh ^Ox dx+ ∫ g/ Z e :Zdx ∫ h/ Yx xdx ∫ k/ H Yx xdx ∫ W, a/ H :Zx dx ∫ b/ dZ xe dx ∫ c/ : ^ x x dx ∫ (x) d/ : H x e xdx ∫ 1<2f'W;'g()5 3 Lê Xuân Manh  §T: 0373840709 , a/ H x e dx + ∫ b/ H /x xdx ∫ c/ E:KYZOc dx ∫ (x) d/ H YK Ox x dx+ ∫ HXHX,,456':MBC',-<+'()*+#  W, a/  ^ _ K Ox x dx+ + ∫ b/ H H    K O e x x dx x x + + + ∫ c/ ^  Hx dx− ∫ d/ H  x dx+ ∫ W, a/ H ^ KH: ^ Ox cosx x dx π π + + ∫ b/  _ K O x e x dx+ ∫ c/  ^ _ K Ox x x dx+ ∫ d/ H  K OK Ox x x dx+ − + ∫ W, a/ H ^  K^: H Ox cosx dx x π π + + ∫ b/  H _ K O x e x dx+ + ∫ c/ H H ^  K Ox x x x dx+ + ∫ d/ H  K OK Ox x x dx− + + ∫ HXHXH,,456'47'4489:; f'B9:; K O K O n(K Op(tKZO Z K O u b b a u a f x d f u du= ∫ ∫ W, a/ H ^ H ^ : xcos xdx π π ∫ b/ H H ^ ^ : xcos xdx π π ∫ c/ H _ :  ^ x dx cosx π + ∫ d/ ` _ tgxdx π ∫ g/ ` s E gxdx π π ∫ h/ s _  `: xcosxdx π + ∫ W, a/  H _ x x dx+ ∫ b/  H _ x x dx− ∫ c/  ^ H _ x x dx+ ∫ d/  H ^ _  x dx x + ∫ 1<2f'W;'g()5 4 Lê Xuân Manh  §T: 0373840709 g/  ^ H _ x x dx− ∫ h/ H ^    dx x x + ∫ W, a/  H _   dx x+ ∫ b/  H   H H dx x x − + + ∫ c/  H _   dx x + ∫ d/  H H _  K ^ O dx x+ ∫ g/ H : ` x e cosxdx π π ∫ h/ H ` : cosx e xdx π π ∫ f/ H  H _ x e xdx + ∫ k/ H ^ H ^ : xcos xdx π π ∫ W, 1/ H : ` x e cosxdx π π ∫ 2/ H ` : cosx e xdx π π ∫ 3/ H  H _ x e xdx + ∫ W, 1/ H ^ H ^ : xcos xdx π π ∫ 2/ H H ^ ^ : xcos xdx π π ∫ 3/ H _ :  ^ x dx cosx π + ∫ 4/ ` _ tgxdx π ∫ 5/ ` s E gxdx π π ∫ 6/ s _  `: xcosxdx π + ∫ W, 1/  H _ x x dx+ ∫ 2/  H _ x x dx− ∫ 3/  ^ H _ x x dx+ ∫ 4/  H ^ _  x dx x + ∫ 5/  ^ H _ x x dx− ∫ 6/ H ^    dx x x + ∫ W, 1/   Y e x dx x + ∫ 2/  :KY O e x dx x ∫ 3/   ^Y Y e x x dx x + ∫ 4/ HY   e x e dx x + ∫ 5/ H H  Y Y e e x dx x x + ∫ 6/ H H  K Y O e e dx cos x+ ∫ W, 1/ H    x dx x+ − ∫ 2/  _ H  x dx x + ∫ 3/  _ x x dx+ ∫ 1<2f'W;'g()5 5 Lê Xuân Manh  §T: 0373840709 4/  _   dx x x+ + ∫ 5/  _   dx x x+ − ∫ 6/ ^  x dx x + ∫  HXHX^,,456'47'44='4! f',45='4!W (K O<tKZO Z K O K O K O tK O b b b a a a x d u x v x v x u x dx= − ∫ ∫ BC D E! D ! F GDH I ! D  J '! F H' u   : K O ax ax f x cosax dx e β α           ∫  K O tK O : : E: ax ax u f x du f x dx ax ax dv ax dx v cosax dx e e = =           ⇒       = =                   ∫ u   K OYK Of x ax dx β α ∫ 1I v  YK O K O K O dx du u ax x dv f x dx v f x dx  = =   ⇒   =   =  ∫ W, / ^ ^  Y e x dx x ∫   Y e x xdx ∫   H _ YK Ox x dx + ∫ B H  Y e x xdx ∫  W, / ^ ^  Y e x dx x ∫   Y e x xdx ∫   H _ YK Ox x dx + ∫ B H  Y e x xdx ∫            !    "#  $%#  $&'  ()  ) aw x B( v KWw x / x w x 45:/( /  H H _ K O x x e dx x + ∫ 8I v  H H K O x u x e dx dv x  =   =  +   ^ y ` ^ H K O x dx x − ∫ 8I v  h ^ ` ^ K O u x x dx dv x  =   =  −       H H H  H H H H H H H H _ _ _ _  K O K O  K O dx x x dx x dx dx I I x x x x + − = = − = − + + + + ∫ ∫ ∫ ∫ 1<2f'W;'g()5 6 Lê Xuân Manh  §T: 0373840709 w x    H _  dx x = + ∫ I z '47'4/ x 48f { * x :f x w x  H [  H H H _ K O x dx x+ ∫ I z '47'4/ x 47 z '45 z W8I v  H H K O u x x dv dx x =    =  +    (  Ww z #Z|_:/EE  H H _  K O x t e dx t = + ∫ 0+S4W,,45:/( / H _ K E:ZO:Zx c dx π + ∫    K OY e x xdx x + ∫  H H  YK Ox x dx + ∫ B ^ H ` /x xdx π π ∫ HX^XP4BC',45,B&,F4R' *!+,-. u&$%94>!L1?8HMN5O(41P&$%89 '!QO&!8&1NRGA!S(41P& $%89 '!QO&!8&1NRGA!S& K O Z b a f x d ∫ uNHMN5O(41P&$%89'!QO&!8 &1 T(KU*VW5$%&<5M K 8 X 8@@@8  )4>!81? T(XU*SHMN5OYN =S&  H  K O Z K O Z] K O Z]XXX K O Z n x x b b a a x x f x d f x d f x d f x d= + ∫ ∫ ∫ ∫ T(ZUB.4>  L [ ?8$%\!)H]@BN K O Z j i x x f x d ∫ W,B&,F4R''bJ} /1~2+#:;)[Z]Z d NDCE+N87•'R'Z[dH<+87•'R'Z[ 1~2+#:;)[e Z ]NDCE+N87•'R'Z[_<+87•'R'Z[ 1~2+#:;)[Z ^ d`ZNDCE+N87•'R'Z[dH<+87•'R'Z[` B1~2+#:;)[:ZNDCE+NDC('<+87•'R'Z[H π W,B&,F4R''bJ} /1~2+#:;)[Z]Z d NDCE+N87•'R'Z[dH<+87•'R'Z[ 1~2+#:;)[e Z ]NDCE+N87•'R'Z[_<+87•'R'Z[ 1~2+#:;)[Z ^ d`ZNDCE+N87•'R'Z[dH<+87•'R'Z[` B1~2+#:;)[:ZNDCE+NDC('<+87•'R'Z[H π uE/+#:;)[\KZON)['KZOY*CD*8EJn/opNB&,F4R''bJ} 1<2f'W;'g()5 7 Lê Xuân Manh  §T: 0373840709 )[\KZON)['KZO<+/87•'R'Z[/NZ[87k,eEf':/( L[ K O K O Z b a f x g x d− ∫ u,B&,F4R''bJ})[\KZON)['KZO<+/87•'R'Z[/NZ[ T(KU*VW5$%&$%5M K 8 X 8@@@8  )4>!81? T(XU*SHMN5OYN =S&  H  K O K O Z K O K O Z] K O K O Z]XXX K O K O Z n x x b b a a x x f x g x d f x g x d f x g x d f x g x d− = − − + − ∫ ∫ ∫ ∫ T(ZUB.4>  L [ ?8$%^$%\!)H]@BN K O K O Z j i x x f x g x d− ∫ HX^XHP4BC',45,Q,<SQ 1<2f'W;'g()5 8