Bài tập Tích phân

BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam NGUYÊN HA ̀ M - TI ́ CH PHÂN VA ̀ Ư ́ NG DU ̣ NG I. NGUYÊN HA ̀ M    ( )   x x dx− + ∫     x x dx x   − +  ÷   ∫  ( )  x x x dx+ + ∫     dx x x   −  ÷   ∫   x dx x −    ÷   ∫  ( ) ( )  x x x dx+ − + ∫  ( )  x x e e dx − − ∫     x x e e dx x −   +  ÷   ∫  ( )  x a x dx+ ∫  ( )   x x dx+ ∫     x x dx x + + ∫  ( )    dx x + ∫    xdx− ∫  ( )   x dx+ ∫  ( )   x x dx+ ∫   x xdx ∫        x x dx   +  ÷   ∫  ( )    x x dx− ∫  ( )    x x dx− ∫  ( )    x x dx+ ∫      dx x x ∫    xdx ∫     dx x x− + ∫      dx x x− + ∫  ( )   x dx+ ∫   xdx ∫   xdx ∫   x x dx+ ∫    x x dx+ ∫    x e xdx ∫    xdx ∫    xdx ∫    xdx ∫    xdx ∫    xdx ∫    xdx ∫  x xe dx ∫   x xe dx ∫  x xdx ∫  x xdx ∫  x xdx ∫   xdx ∫  x xdx ∫   x xdx ∫    x dx x + ∫      x e xdx ∫     x dx x + ∫    x dx x + ∫ II. TI ́ CH PHÂN --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam      xdx ∫   e dx x ∫      dx x ∫         x dx x   +  ÷   ∫      x x dx x − ∫       e x x dx x + − ∫      xdx π ∫      xdx π ∫     x xdx π π − ∫       x xdx π π − ∫       x x dx x − + + ∫          x x dx x − + − ∫       x dx π π   −  ÷   ∫      dx x x+ + − ∫  ( )    x cos x dx π π − + ∫  ( )     x cos x dx π π − ∫      xdx π ∫      xdx π ∫      xdx π ∫      xdx π ∫     x dx− ∫      x x dx+ − ∫    x dx π ∫      o xdx π − ∫     x xe dx ∫     x xe dx − ∫  ( )     x dx+ ∫      xdx x + ∫      x x dx− ∫      xdx π ∫      xdx π ∫     xdx π ∫ --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam     xdx π π ∫       x dx x π + ∫      x xdx π + ∫      x e xdx π ∫     e x dx x + ∫        x e xdx π ∫       e x dx x x− ∫       x dx x + ∫      x x dx+ ∫      x xdx π ∫        x dx x π + ∫       x x dx+ ∫      a dx a x− ∫    x a t =       dx x− ∫      dx x− ∫      dx x− ∫      a a x dx− ∫    x a t =      x dx− ∫      x dx− ∫      x dx− ∫     a dx a x+ ∫    x a t=      dx x+ ∫      dx x x+ + ∫      x x dx− ∫         dx x x− ∫     π π π   = −  ÷         x dx x + − ∫    x t=     dx x π π ∫         !    x t t t x x t t − = ⇒ = = + +       x dx x + ∫         x dx x − − ∫    x x x e dx e e − + ∫     x dx e − ∫         x dx x x + + − ∫ --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam       xdx x x+ + ∫       dx x x π π ∫  ( )    e x dx x π ∫  ( )     e x dx x π ∫     x xe dx ∫  ( )    x xdx π − ∫  ( )     x xdx π − ∫     x x e dx − ∫     x xdx π ∫     x xdx π ∫     x e xdx π ∫     x e xdx π ∫    e x xdx ∫    e xdx ∫     e xdx ∫  ( )     x x dx− ∫     e x dx x ∫     e x xdx ∫     x xdx ∫     x xdx ∫    x xdx ∫     x xdx ∫ III. BÂ ́ T ĐĂ ̉ NG THƯ ́ C CU ̉ A TI ́ CHPHÂN "#  $% &  :          x dx + ≤ ≤ ∫          dx x − ≤ ≤ + ∫         x x dx − ≤ ≤ ∫          dx x π π π π ≤ ≤ − ∫  ( )       x x dx − ≤ + + − ≤ ∫          xdx π π π π ≤ + ≤ ∫        dx x π π π ≤ ≤ + ∫  --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam IV. ÔN TÂ ̣ P VÊ ̀ TI ́ CH PHÂN '( ∫ + + =     π dx x xx I '( dx x xx I ∫ + =     π '( ( ) ∫ +=     π xdxxeI x '( dx x x I ∫ + + =      '(     I x xdx π = ∫ '( ( )      x I x e x dx π = + ∫ '( ∫ = e xdxxI    '( dxxxI ∫ +=     '( ∫ − +++ − =     dx xx x I '( dxxxI ∫ −=     '( ∫ =     π xdxeI x '(     I x x dx= + ∫ '( ∫ + − =      π dx x x I '( ∫ − ++ =    xx dx I '( ∫ = e dx x x I    '( dx x x I ∫ + + =       '( ∫ + =     π dx x x I '( ∫ ∫ = + =             π π xx xdxx J x xx xdx I '( ∫ = e xdxxI   '( dxxxI     ∫ = π '( dx x xxx I ∫ + +++ =       '( ( ) ∫ + =    x xdx I '( ∫ − = e xx dx I    '( ∫ + =       π dx xx x I '( ∫ + =      π dx x x I '( 2 2 2 0 sin 2x I dx cos x 4sin x π = + ∫ '( 6 2 dx I 2x 1 4x 1 = + + + ∫ '( ( ) 1 2x 0 I x 2 e dx= − ∫ --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam '( ( ) 2 0 I x 1 sin 2x dx π = + ∫ '( ( ) 2 1 I x 2 ln x dx= − ∫ '( ln5 x x ln3 dx I e 2e 3 − = + − ∫ '( 10 5 dx I x 2 x 1 = − − ∫ '( e 1 3 2 ln x I dx x 1 2 ln x − = + ∫ '( ( ) 1 2 0 I x ln 1 x dx= + ∫  )*+,-. 2 t 1 x= + /012) '( ( ) 2 2 1 ln 1 x I dx x + = ∫ '( 1 2 0 I x x 1dx= + ∫ '( 1 2 0 x I dx 1 x = + ∫ '( 2 4 sin x cosx I dx 1 sin2x π π − = + ∫ '( ( ) 3 2 0 I x ln x 5 dx = + ∫ '( ( ) 2 3 0 cos2x I dx sin x cos x 3 π = − + ∫ '( ( ) 4 0 I x 1 cosx dx π = − ∫ '( 4 0 cos2x I dx 1 2sin 2x π = + ∫ '( ln2 2x x 0 e I dx e 2 = + ∫ '( 3 2 0 4sin x I dx 1 cos x π = + ∫ '( 4 2 0 x I dx cos x π = ∫ '( 3 1 x 3 I dx 3 x 1 x 3 − − = + + + ∫ '( 9 3 1 I x. 1 x dx= − ∫ '( e 3 1 x 1 I ln x dx x   + =  ÷   ∫ '( 1 2 3 0 I x 2 x dx= + ∫ '( ( ) ∫ −=     π xdxxI '( ( ) ∫ −+=      dxxexI x '( = − ∫ ln 4 x ln2 dx I e 1 '( ( ) 1 2 0 I x ln 1 x dx = + ∫ '( 2 1 x x 1 I dx x 5 − = − ∫ '( ( ) π = + ∫ 2 3 0 I x cos x sin x dx '( 2 0 cosx I dx 5 2sin x π = − ∫ '( ( ) ( ) = + + ∫ 2 0 I 2x 7 ln x 1 dx '( ( ) π = − ∫ 4 8 0 I 1 tan x dx '( 4 2 3 4x 3 I dx x 3x 2 + = − + ∫ '( 3 6 0 sin 3x sin 3x I dx 1 cos3x π − = + ∫ --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam '( e 3 2 1 ln x 2 ln x I dx x + = ∫ '( ( ) 4 4 4 0 I cos x sin x dx π = − ∫ '( 4 0 cos2x I dx 1 2sin 2x π = + ∫ '( 2 0 I sin x sin 2xdx π = ∫ '( ( ) 1 2 0 x I dx x 3 = + ∫ '( π = ∫ 2 2 0 I x cos xdx '( ( ) e 2 1 dx I x 1 ln x = + ∫ '( 2 4 sin x cosx I dx 1 sin 2x π π − = + ∫ '( ( ) π π = ∫ 3 4 ln tan x I dx sin 2x '( ( ) 2 3 2 0 I sin 2x 1 sin x dx π = + ∫ '( e 0 ln x I dx x = ∫ '( 1 2 0 1 I dx x 2x 2 = + + ∫ '( = − ∫ 2 2 2 2 0 x I dx 1 x '( 4 2 0 x I dx cos x π = ∫ '( ( ) 2 1 I 4x 1 ln x dx = − ∫ '( 3 6 dx I sin x.sin x 3 π π π =   +  ÷   ∫ '( e 3 2 1 I x ln x dx= ∫ '( 4 0 2x 1 dx 1 2x 1 + + + ∫ '( ( ) 1 2 0 x x 1 dx x 4 − − ∫ '( 2 2 0 x cosxdx π ∫ '( 3 2 0 4 cos x dx 1 sin x π + ∫ '( 7 3 0 x 2 dx x 1 + + ∫ '( 2007 1 2 1 3 1 1 1 dx x x   +  ÷   ∫ '( ( ) e 2 1 x ln x dx ∫ '( ( ) 4 2 1 x sin x dx π ∫ '(    x dx− ∫ '( ( ) 3 2 2 1 dx x x 1+ ∫ '( 3 3 2 1 x x 1dx− ∫ '( ( ) 0 2x 1 x e x 1 dx − + + ∫ '(      x dx x π ∫ --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam '( ( )           x dx x x x π π   −  ÷   + + + ∫ '(     x dx x ∫ V. Ư ́ NG DU ̣ NG CU ̉ A TI ́ CH PHÂN '(1. 345361789-:;#< ( ) ( ) x y e 1 x, y 1 e x= + = + '( "617)89-:;#< y x ln x= / y 0, y e= = 3=3>?@%ABC9 (?DEC6)DE%FGB '( 345361789-:;#< ( )    (  − = = + x x y v y x  '( 345361789-:;#<   ( = = −y x v y x . '( 345361789-:;#<H1#I%6 2 y x 2= − ! y x; x 1; x 0= = − =  '( 345361789-:;#< y x = / 2 y x cos x= + / x 0 = / x = π  '( 345361789-:1%- ( )   P y x x= − + J(#<7 d y x=  --------------------------------------------------------------------------------------------------------------------------------------------------------- Năm ho ̣ c 2010 - 2011 . BA ̀ I TÂ ̣ P TOA ́ N LƠ ́ P 12 Nguyễn Thanh Lam NGUYÊN HA ̀ M - TI ́ CH PHÂN VA ̀ Ư ́ NG DU ̣ NG I. NGUYÊN HA ̀ M    ( )   x x dx− +.     x e xdx ∫     x dx x + ∫    x dx x + ∫ II. TI ́ CH PHÂN ---------------------------------------------------------------------------------------------------------------------------------------------------------