KỸ THUẬT TÍNH TÍCH PHÂN pdf

TÍNH CÁC TÍCH PHÂN SAU III. TÍCH PHÂN HÀM HỮU TỶ:    5 3 2 23 12 dx xx x   b a dx bxax ))(( 1    1 0 3 1 1 dx x xx dx x xx    1 0 2 3 1 1   1 0 3 2 )13( dx x x   1 0 22 )3()2( 1 dx xx    2 1 2008 2008 )1( 1 dx xx x     0 1 2 23 23 9962 dx xx xxx   3 2 22 4 )1( dx x x    1 0 2 32 )1( dx x x n n    2 1 24 2 )23( 3 dx xxx x   2 1 4 )1( 1 dx xx   2 0 2 4 1 dx x   1 0 4 1 dx x x dx xx   2 0 2 22 1   1 0 32 )1( dx x x   4 2 23 2 1 dx xxx    3 2 3 2 23 333 dx xx xx    2 1 4 2 1 1 dx x x   1 0 3 1 1 dx x    1 0 6 456 1 2 dx x xxx    1 0 2 4 1 2 dx x x    1 0 6 4 1 1 dx x x IV. TÍCH PHÂN HÀM LƯỢNG GIÁC: xdxx 4 2 0 2 cossin    2 0 32 cossin  xdxx dxxx  2 0 54 cossin    2 0 33 )cos(sin  dxx   2 0 44 )cos(sin2cos  dxxxx   2 0 22 )coscossinsin2(  dxxxxx  2 3 sin 1   dx x   2 0 441010 )sincoscos(sin  dxxxxx   2 0 cos2  x dx   2 0 sin2 1  dx x   2 0 2 3 cos1 sin  dx x x  3 6 4 cos.sin   xx dx   4 0 22 coscossin2sin  xxxx dx   2 0 cos1 cos  dx x x   2 0 cos2 cos  dx x x   2 0 sin2 sin  dx x x   2 0 3 cos1 cos  dx x x   2 0 1cossin 1  dx xx   2 3 2 )cos1( cos   x xdx     2 2 3cos2sin 1cossin   dx xx xx  4 0 3  xdxtg dxxg  4 6 3 cot    3 4 4   xdxtg   4 0 1 1  dx tgx   4 0 ) 4 cos(cos   xx dx    2 0 5cos5sin4 6cos7sin  dx xx xx    2 0 sin1 dxx   4 0 13cos3sin2  xx dx   4 0 4 3 cos1 sin4  dx x x    2 0 cossin 2sin2cos1  dx xx xx   2 0 cos1 3sin  dx x x   2 4 sin2sin   xx dx  4 0 2 3 cos sin  dx x x   2 0 32 )sin1(2sin  dxxx   0 sincos dxxx   3 4 3 3 3 sin sinsin   dx xtgx xx   2 0 cossin1  xx dx   2 0 1sin2  x dx  2 4 53 sincos   xdxx   4 0 2 cos1 4sin  x xdx   2 0 3sin5  x dx  6 6 4 cossin   xx dx   3 6 ) 6 sin(sin    xx dx   3 4 ) 4 cos(sin    xx dx  3 4 6 2 cos sin   x xdx dxxtgxtg ) 6 ( 3 6        3 0 3 )cos(sin sin4  xx xdx    0 2 2 )sin2( 2sin  x x  2 0 3 sin  dxx  2 0 2 cos  xdxx   2 0 12 .2sin  dxex x dxe x x x    2 0 cos1 sin1    4 6 2cot 4sin3sin   dx xgtgx xx   2 0 2 6sin5sin 2sin  xx xdx dxxx   2 0 2 cos)12(    0 2 cossin xdxxx  4 0 2  xdxxtg   0 22 sin xdxe x  2 0 3sin cossin 2  xdxxe x   4 0 )1ln(  dxtgx   4 0 2 )cos2(sin  xx dx    2 0 2 )cos2)(sin1( cos)sin1(  dx xx xx V. TÍCH PHÂN HÀM VÔ TỶ:  b a dxxfxR ))(,( Trong ®ã R(x, f(x)) cã c¸c d¹ng: +) R(x, xa xa   ) §Æt x = a cos2t, t ] 2 ;0[   +) R(x, 22 xa  ) §Æt x = ta sin hoÆc x = ta cos +) R(x, n dcx bax   ) §Æt t = n dcx bax   +) R(x, f(x)) =   xxbax 2 )( 1 Víi (   xx 2 )’ = k(ax+b) Khi ®ã ®Æt t =   xx 2 , hoÆc ®Æt t = b ax  1 +) R(x, 22 xa  ) §Æt x = tgta , t ] 2 ; 2 [    +) R(x, 22 ax  ) §Æt x = x a cos , t } 2 {\];0[    +) R   1 2 i n n n x x x ; ; ; Gäi k = BCNH(n 1 ; n 2 ; ; n i ) §Æt x = t k 1.   32 5 2 4xx dx 2.   2 3 2 2 1xx dx 3.    2 1 2 1 2 5124)32( xxx dx 4.   2 1 3 1xx dx 5.   2 1 2 2008dxx 6.   2 1 2 2008x dx 7.   1 0 22 1 dxxx 8.   1 0 32 )1( dxx 9.    3 1 22 2 1 1 dx xx x 10.    2 2 0 1 1 dx x x 11.   1 0 32 )1( x dx 12.   2 2 0 32 )1( x dx 13.   1 0 2 1 dxx 14.   2 2 0 2 2 1 x dxx 15.   2 0 2cos7 cos  x xdx 16.   2 0 2 coscossin  dxxxx 17.   2 0 2 cos2 cos  x xdx 18.    2 0 cos31 sin2sin  dx x xx 19.   7 0 3 2 3 1 x dxx 20.   3 0 23 10 dxxx 21.   1 0 12x xdx 22.   1 0 2 3 1xx dxx 23.   7 2 112x dx 24. dxxx   1 0 815 31 25. 2 0 56 3 cossincos1 xdxxx 26. 3ln 0 1 x e dx 27. 1 1 2 11 xx dx 28. 2ln 0 2 1 x x e dxe 29. 1 4 5 2 8412 dxxx 30. e dx x xx 1 lnln31 31. 3 0 2 35 1 dx x xx 32. dxxxx 4 0 23 2 33. 0 1 3 2 )1( dxxex x 34. 3ln 2ln 2 1ln ln dx xx x 35. 3 0 2 2 cos 32 cos 2cos dx x tgx x x 36. 2ln 0 3 )1( x x e dxe 37. 3 0 2cos2 cos x xdx 38. 2 0 2 cos1 cos x xdx 39. dx x x 7 0 3 3 2 40. a dxax 2 0 22 VI. MT S TCH PHN C BIT: Bài toán mở đầu: Hàm số f(x) liên tục trên [-a; a], khi đó: aa a dxxfxfdxxf 0 )]()([)( Ví dụ: +) Cho f(x) liên tục trên [- 2 3 ; 2 3 ] thỏa mãn f(x) + f(-x) = x2cos22 , Tính: 2 3 2 3 )( dxxf +) Tính 1 1 2 4 1 sin dx x xx Bài toán 1: Hàm số y = f(x) liên tục và lẻ trên [-a, a], khi ®ã:   a a dxxf )( = 0. VÝ dô: TÝnh:    1 1 2 )1ln( dxxx    2 2 2 )1ln(cos   dxxxx Bµi to¸n 2: Hµm sè y = f(x) liªn tôc vµ ch½n trªn [-a, a], khi ®ã:   a a dxxf )( = 2  a dxxf 0 )( VÝ dô: TÝnh    1 1 24 1xx dxx 2 2 2 cos 4 sin     x x dx x   Bµi to¸n 3: Cho hµm sè y = f(x) liªn tôc, ch½n trªn [- a, a], khi ®ã:     aa a x dxxfdx b xf 0 )( 1 )( (1  b>0,  a) VÝ dô: TÝnh:     3 3 2 21 1 dx x x    2 2 1 5cos3sinsin   dx e xxx x Bµi to¸n 4: NÕu y = f(x) liªn tôc trªn [0; 2  ], th×   2 0 2 0 )(cos)(sin  dxxfxf VÝ dô: TÝnh   2 0 20092009 2009 cossin sin  dx xx x   2 0 cossin sin  dx xx x Bµi to¸n 5: Cho f(x) x¸c ®Þnh trªn [-1; 1], khi ®ã:     00 )(sin 2 )(sin dxxfdxxxf VÝ dô: TÝnh    0 sin1 dx x x    0 cos2 sin dx x xx Bµi to¸n 6:   b a b a dxxfdxxbaf )()(    bb dxxfdxxbf 00 )()( VÝ dô: TÝnh    0 2 cos1 sin dx x xx   4 0 )1ln(4sin  dxtgxx Bµi to¸n 7: NÕu f(x) liªn tôc trªn R vµ tuÇn hoµn víi chu k× T th×:    TTa a dxxfdxxf 0 )()(    TnT dxxfndxxf 00 )()( VÝ dô: TÝnh    2008 0 2cos1 dxx C¸c bµi tËp ¸p dông: 1.     1 1 2 21 1 dx x x 2.    4 4 4 357 cos 1   dx x xxxx 3.    1 1 2 )1)(1( xe dx x 4.     2 2 2 sin4 cos   dx x xx 5.     2 1 2 1 ) 1 1 ln(2cos dx x x x 6. dxnx)xsin(sin 2 0    7.    2 2 5 cos1 sin   dx x x 8. 1 )1(1 cot 1 2 1 2      ga e tga e xx dx x xdx (tana>0) VII. TÍCH PHÂN HÀM GIÁ TRỊ TUYỆT ĐỐI: 1.    3 3 2 1dxx 2.   2 0 2 34 dxxx 3.   2 0 2 dxxx 4.   2 2 sin   dxx 5.      dxxsin1 6.   3 6 22 2cot   dxxgxtg 7.  4 3 4 2sin   dxx 8.    2 0 cos1 dxx 9.    5 2 )22( dxxx 10.   3 0 42 dx x 11.    3 2 3 coscoscos   dxxxx . TÍNH CÁC TÍCH PHÂN SAU III. TÍCH PHÂN HÀM HỮU TỶ:    5 3 2 23 12 dx xx x   b a dx bxax ))(( 1    1 0 3 1 1 dx x xx .    1 0 6 456 1 2 dx x xxx    1 0 2 4 1 2 dx x x    1 0 6 4 1 1 dx x x IV. TÍCH PHÂN HÀM LƯỢNG GIÁC: xdxx 4 2 0 2 cossin    2 0 32 cossin  xdxx dxxx  2 0 54 cossin  .   4 0 )1ln(  dxtgx   4 0 2 )cos2(sin  xx dx    2 0 2 )cos2)(sin1( cos)sin1(  dx xx xx V. TÍCH PHÂN HÀM VÔ TỶ:  b a dxxfxR ))(,( Trong ®ã R(x, f(x)) cã c¸c d¹ng: +) R(x, xa xa   )