I. TÍNH TÍCH PHÂN BẰNG CÁCH SỬ DỤNG TÍNH CHẤT VÀ NGUYÊN HÀM CƠ BẢN:      x x dx+ + ∫ 2.        e x x dx x x + + + ∫ 2.   x dx− ∫ 3.   x dx+ ∫ 4.     x cosx x dx π π + + ∫ 5.     x e x dx+ ∫ 6.     x x x dx+ ∫ 7.     x x x dx+ − + ∫ 8.      x cosx dx x π π + + ∫ 9.      x e x dx+ + ∫ 10.      x x x x dx+ + ∫ 11.     x x x dx− + + ∫ 12.      ( ). − + ∫ 13.  2 2 -1 x.dx x + ∫ 14.          − − ∫ 15.   5 2 dx x 2+ + − ∫ 16.          ( ). ln + + ∫ 17.        cos . sin π π ∫ 18.       . cos π ∫ 19.           dx − − − + ∫ 20.          . − + ∫ 21.      + ∫ 22.        ln . − + ∫ 22.     sin π + ∫ II. PHƯƠNG PHÁP ĐẶT ẨN PHỤ:  1.      xcos xdx π π ∫ 2.      xcos xdx π π ∫ 3.      x dx cosx π + ∫ 3.   tgxdx π ∫ 4.    gxdx π π ∫ 5.     xcosxdx π + ∫ 6.    x x dx+ ∫ 7.    x x dx− ∫ 8.     x x dx+ ∫ 9.      x dx x + ∫       x x dx− ∫        dx x x + ∫       dx x+ ∫         dx x x − + + ∫        dx x + ∫           dx x+ ∫      x e cosxdx π π ∫      cosx e xdx π π ∫  18.     x e xdx + ∫ 19.      xcos xdx π π ∫ 20.    x e cosxdx π π ∫ 21.    cosx e xdx π π ∫ 22.     x e xdx + ∫        xcos xdx π π ∫        xcos xdx π π ∫        x dx cosx π + ∫     tgxdx π ∫      gxdx π π ∫       xcosxdx π + ∫     x x dx+ ∫ 30.    x x dx− ∫ 31.     x x dx+ ∫ 32.      x dx x + ∫ 33.     x x dx− ∫ 34.      dx x x + ∫ 35.    e x dx x + ∫ 36.    e x dx x ∫ 37.     e x x dx x + ∫ 38.    e x e dx x + ∫ 39.      e e x dx x x + ∫ 40.       e e dx cos x+ ∫ 41.     x dx x+ − ∫ 42.     x dx x + ∫ 43.   x x dx+ ∫ 44.     dx x x+ + ∫ 45.     dx x x+ − ∫ 46.   x dx x + ∫      e x dx x + ∫ 47.    e x dx x ∫ 48.     e x x dx x + ∫ 49.    e x e dx x + ∫ 50.      e e x dx x x + ∫ 51.       e e dx cos x+ ∫ 52.     + ∫ x x dx 53. ( )      + ∫ x xdx π 54.     x dx− ∫ 55.     x dx− ∫ 56.     dx x+ ∫ II. PHƯƠNG PHÁP TÍCH PHÂN TỪNG PHẦN:  !"# $ %&        &  b b b a a a x d u x v x v x u x dx= − ∫ ∫                'Da ̣ ng 1    ax ax f x cosax dx e β α           ∫    &     ax ax u f x du f x dx ax ax dv ax dx v cosax dx e e = =           ⇒       = =                   ∫ 'Da ̣ ng 2:   f x ax dx β α ∫ () *        dx du u ax x dv f x dx v f x dx  = =   ⇒   =   =  ∫ 'Da ̣ ng 3:         ∫ ax ax e dx cosax β α  +, - $ * #, - . - , -  .$ ./       x x e dx x + ∫ 0) *      x u x e dx dv x  =   =  +  1/        x dx x − ∫ 0) *        u x x dx dv x  =   =  −  /                             dx x x dx x dx dx I I x x x x + − = = − = − + + + + ∫ ∫ ∫ ∫ 2, - 3      dx x = + ∫ 1) 4 56. - 0 7 18 -  - 2, - 3  9        x dx x+ ∫ 1) 4 56. - 5 4  4 #0) *      u x x dv dx x =    =  +  Bài tập      e x dx x ∫     e x xdx ∫      x x dx + ∫      e x xdx ∫       e x dx x ∫     e x xdx ∫       x x dx + ∫      e x xdx ∫     x c dx π + ∫       e x xdx x + ∫       x x dx + ∫      .x xdx π π ∫ 13.     x dx x ∫ 14.   x xdx π ∫ 15.   x xe dx ∫ 16.    x e xdx π ∫ III. TÍCH PHÂN HÀM HỮU TỶ:  ∫ +− −      dx xx x  ∫ ++ b a dx bxax    ∫ + ++      dx x xx  dx x xx ∫ + ++        ∫ +      dx x x  ∫ ++      dx xx  ∫ + −       dx xx x  ∫ − +− ++−       dx xx xxx  ∫ −      dx x x  ∫ + −      dx x x n n  ∫ ++ −       dx xxx x  ∫ +      dx xx  ∫ +      dx x  ∫ +     dx x x  dx xx ∫ +−       ∫ +     dx x x  ∫ +−      dx xxx  ∫ +− ++       dx xx xx  ∫ + −       dx x x  ∫ +      dx x  ∫ + +++       dx x xxx  ∫ + −       dx x x  ∫ + +       dx x x         x dx x x + + + ∫      dx x x+ + ∫                IV. TÍCH PHÂN HÀM LƯỢNG GIÁC:  xdxx      ∫ π  ∫     π xdxx  dxxx ∫     π  ∫ +     π dxx  ∫ +     π dxxxx  ∫ −−     π dxxxxx  ∫     π π dx x  ∫ −+     π dxxxxx  ∫ −    π x dx  ∫ +     π dx x  ∫ +       π dx x x  ∫     π π xx dx  ∫ −+     π xxxx dx  ∫ +     π dx x x  ∫ −     π dx x x  ∫ +     π dx x x  ∫ +      π dx x x  ∫ ++     π dx xx  ∫ −      π π x xdx  ∫ − ++ +−     π π dx xx xx  ∫    π xdxtg  dxxg ∫     π π  ∫    π π xdxtg  ∫ +     π dx tgx  ∫ +      π π xx dx  ∫ ++ ++     π dx xx xx  ∫ + π    dxx  ∫ ++    π xx dx  ∫ +       π dx x x  ∫ + ++     π dx xx xx  ∫ +     π dx x x  ∫ −    π π xx dx  ∫       π dx x x  ∫ +     π dxxx  ∫ π   dxxx  ∫ −        π π dx xtgx xx  ∫ ++    π xx dx  ∫ +    π x dx  ∫     π π xdxx  ∫ +      π x xdx  ∫ +    π x dx  ∫     π π xx dx  ∫ +      π π π xx dx  ∫ +      π π π xx dx  ∫       π π x xdx  dxxtgxtg      π π π ∫ +  ∫ +      π xx xdx  ∫ − +      π x x  ∫     π dxx  ∫     π xdxx  ∫ +     π dxex x  dxe x x x ∫ + +     π  ∫ +     π π dx xgtgx xx  ∫ +−      π xx xdx  ∫    dxx  ∫      π π dx x x  dxxx ∫ −     π  ∫ π    xdxxx  ∫    π xdxxtg  ∫ π    xdxe x  ∫      π xdxxe x  ∫ +    π dxtgx  ∫ +     π xx dx  ∫ −+ −      π dx xx xx        − ∫ x xdx π π          + ∫ x x x dx π        + ∫ x dx x π              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 ;  <= π ∈ +) R(x,  xa − ) §Æt x = ta  hoÆc x = ta  +) R(x, n dcx bax + + ) §Æt t = n dcx bax + + +) R(x, f(x)) = γβα +++ xxbax    Víi ( γβα ++ xx  )’ = k(ax+b) Khi ®ã ®Æt t = γβα ++ xx  , hoÆc ®Æt t = bax +  +) R(x,  xa + ) §Æt x = tgta , t ;  <  = ππ −∈ +) R(x,  ax − ) §Æt x = x a  , t >  ?@;<= π π ∈ +) R ( )         ; ; .; Gäi k = BCNH(n 1 ; n 2 ; .; n i ) §Æt x = t k  ∫ +    xx dx 2. ∫ −     xx dx 3. ∫ − +++       xxx dx 4. ∫ +    xx dx 5. ∫ +    dxx 6. ∫ +    x dx 7. ∫ +     dxxx 8. ∫ −     dxx 9. ∫ + +       dx xx x 10. ∫ − +      dx x x 11. ∫ +     x dx 12. ∫ −      x dx 13. ∫ +     dxx 14. ∫ −       x dxx 15. ∫ +     π x xdx 16. ∫ −     π dxxxx 17. ∫ +      π x xdx 18. ∫ + +     π dx x xx 19. ∫ +       x dxx 20. ∫ −     dxxx 21. ∫ +   x xdx 22. ∫ ++     xx dxx 23. ∫ ++   x dx 24. dxxx ∫ +     25. ∫ −      π xdxxx 26. ∫ +    x e dx 27. ∫ − +++     xx dx 28. ∫ +     x x e dxe 29. ∫ −−      dxxx 30. ∫ + e dx x xx   31. ∫ + +      dx x xx 32. dxxxx ∫ +−     33. ∫ − ++      dxxex x 34. ∫ +      dx xx x 35. + dx x tgx x x 36. + x x e dxe 37. + x xdx 38. + x xdx 39. dx x x + + 40. + a dxax 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 ;= Ví dụ: +) Cho f(x) liên tục trên [- < ] thỏa mãn f(x) + f(-x) = x , Tính: dxxf +) Tính + + dx x xx Bài toán 1: Hàm số y = f(x) liên tục và lẻ trên [-a, a], khi đó: a a dxxf = 0. Ví dụ: Tính: ++ dxxx ++ dxxxx Bài toán 2: Hàm số y = f(x) liên tục và chẵn trên [-a, a], khi đó: a a dxxf = 2 a dxxf Ví dụ: Tính + xx dxx + x x dx x Bài toán 3: Cho hàm số y = f(x) liên tục, chẵn trên [-a, a], khi đó: = + aa a x dxxfdx b xf (1 b>0, a) Ví dụ: Tính: + + dx x x + dx e xxx x Bài toán 4: Nếu y = f(x) liên tục trên [0; ], thì = dxxfxf [...]... phng gii hn bi a/ th hm s y = x + x -1 , trc honh , ng thng x = -2 v ng thng x =1 b/ th hm s y = ex +1 , trc honh , ng thng x = 0 v ng thng x = 1 c/ th hm s y = x3 - 4x , trc honh , ng thng x = -2 v ng thng x =4 d/ th hm s y = sinx , trc honh , trc tung v ng thng x = 2 Vớ d 2 : Tớnh din tớch hỡnh phng gii hn bi a/ th hm s y = x + x -1 , trc honh , ng thng x = -2 v ng thng x =1 b/ th hm s y = ex... cos 2009 x 0 sin sin x sin x + cos x 0 xf (sin x)dx = Bài toán 5: Cho f(x) xác định trên [-1 ; 1], khi đó: 0 x 1 + sin x dx Ví dụ: Tính Bài toán 6: a f ( a + b x )dx = f ( x )dx b a Ví dụ: Tính 2 0 0 x sin x 0 b f (b x ) dx = f ( x ) dx 0 x sin x 1 + cos f (sin x)dx 0 b 2 + cos x dx 0 b 2 dx 4 dx x sin 4 x ln(1 + tgx)dx 0 Bài toán 7: Nếu f(x) liên tục trên R và tuần hoàn với chu... cos x dx 0 b 2 dx 4 dx x sin 4 x ln(1 + tgx)dx 0 Bài toán 7: Nếu f(x) liên tục trên R và tuần hoàn với chu kì T thì: a +T a T f ( x )dx = f ( x )dx 0 Ví dụ: Tính 2008 nT 0 1 cos 2 x dx 0 Các bài tập áp dụng: 1 1 1 x dx 1+ 2x 1 1 3 4 2 (1 + e 1 x 2 dx )(1 + x 2 ) 4 2 2 6 sin(sin x + nx)dx 0 2 7 x + cos x dx 2 x 4 sin 1 x 1cos 2 x ln(1 + x )dx x7 x5 + x3 x + 1 dx cos 4 x 4 2 1 2 5... Tớnh din tớch hỡnh phng gii hn bi a/ th hm s y = x + x -1 , trc honh , ng thng x = -2 v ng thng x =1 b/ th hm s y = ex +1 , trc honh , ng thng x = 0 v ng thng x = 1 c/ th hm s y = x3 - 4x , trc honh , ng thng x = -2 v ng thng x =4 d/ th hm s y = sinx , trc honh , trc tung v ng thng x = 2 TNH TH TCH VT TH TRềN XOAY . 2, - 3      dx x = + ∫ 1) 4 56. - 0 7 18 -  - 2, - 3  9        x dx x+ ∫ 1) 4 56. - 5 4. 'Da ̣ ng 3:         ∫ ax ax e dx cosax β α  +, - $ * #, - . - , -  .$ ./       x x e dx x + ∫ 0)