Hocmai.vn – Website h c tr c n s t i Vi t Nam Khóa h c Luy n thi THPT qu c gia PEN-C: Môn Toán (Th y Nguy n Bá Tu n) Nguyên hàm – tích phân TÍCH PHÂN HÀM PHÂN TH C ÁP ÁN BÀI T P T LUY N Giáo viên: NGUY N BÁ TU N Các t p tài li u đ c biên so n kèm theo gi ng Tích phân hàm phân th c thu c khóa h c Luy n thi THPT qu c gia PEN-C: Môn Toán (Th y Nguy n Bá Tu n – Phan Huy Kh i – Tr n Ph s d ng hi u qu , B n c n h c tr ng) t i website Hocmai.vn c Bài gi ng sau làm đ y đ t p tài li u Bài 1: Tìm h nguyên hàm sau: dx  4x  c  dx x  3x  a x a x b  9x c x d  4x dx  12 x  d  dx x  3x  Gi i b 2 1 C dx   dx    4x  x  x  2 2  9x 1 1 1 dx   dx   dx   C 2  9x   12 x   9 2 2  9 x    x   x  3  3 3   1 1 x C dx   dx   dx   dx  ln x   ln x   C  ln  3x  x x 1 x 1  x  1 x   1 1  dx   dx    dx   dx   3x   x 1 4x 1   x  1 x  1 x 1 1 C ln x   ln x    C  ln  5 4x 1 Bài 2: Tìm h nguyên hàm sau:  x   dx  4x  2x  d  dx x  4x  Gi i 2( x  1) dx  2x  3x  c  dx x  2x  a x b x d  x2  x  3 2( x  1) 2x  a   ln x2  x   C dx   dx   x  2x  x  2x  x  2x  b d  x2  x  3  x   dx x  4dx    x2  x   x2  x   x2  x   ln x  x   C c Ta có : +) A x  3  B  x  1  A B x  A B 3x  3x  A B      x  x   x  1 x  3 x  x   x  1 x  3  x  1 x  3 Hocmai.vn – Ngôi tr ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Hocmai.vn – Website h c tr c n s t i Vi t Nam Khóa h c Luy n thi THPT qu c gia PEN-C: Môn Toán (Th y Nguy n Bá Tu n) Nguyên hàm – tích phân  A    A B   ng nh t h s hai t s ta có h :   3 A B   B   Suy : 3x    x  x   x  1  x  3 3x  7 dx   dx   dx  ln x   ln x   C  2x  x 1 x3 4 E  x    D Ex  D  E 2x  ng nh t h s hai t s : d.Ta có :   x  4x  x2  x  x  4x  2 E  E  Ta có h     D  E  3  D  7 2x  2x  Suy :   x  4x  x  4x  x  4x  2x  2x  V y:  dx   dx   dx  ln x2  x   C x  4x  x  4x  x  x  2 x V y: Bài Tính tích phân sau a  1 d dx x  x3 x  1  x b e dx x3dx 0 x2  x  x2 dx  1  x c dx 2 x  x  1 h  x3  x2  x  k  dx x2  3x  1  x  11 dx x2  x  3x2  3x  l  dx x  3x  2 dx  x 1  x f g dx 0 x2  5x  x3  x  0 x  dx i m x2   3x  1 dx Gi i a dx  x x   Phân tích : f ( x)   A B   A  1 x     B  1  f ( x)   ng nh t h s hai t s ta có : C  x  x2 A C     A Bx  C  A B x  Cx  A 1     x  x3 x 1  x2  x  x2 x 1  x2    f ( x)dx  Hocmai.vn – Ngôi tr 1 x    ln  ln  dx   ln x  ln 1  x2    2      x   x ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Hocmai.vn – Website h c tr c n s t i Vi t Nam Khóa h c Luy n thi THPT qu c gia PEN-C: Môn Toán (Th y Nguy n Bá Tu n) b x  V y: c x  1 0  d  1   f ( x)dx    x   x   dx   ln x   ln x    2ln  ln x dx  2x 1 Phân tích : f ( x)  5x   2x   x3  x 2  x 2   2  x  x    x  12 x  2x  x  2x    2x    V y : I   f ( x)dx    x      dx    x  x    x  12  0 3 1   x2  x  ln  x  1     10ln  x  1 2  1 1    x  x   x   x  3 x  x  2 dx  5x  Phân tích : f ( x)   Nguyên hàm – tích phân x  1  x 3 dx  Phân tích : f ( x)  x 4Cx2   B  4C  x  A B  C C     3 1  x 1  x  x 1  x A 1  x B  A  4C   1    f ( x)     ng nh t h s hai t s : 2 B  4C    B  2 1  x 1  x  A B  C    C    1   1 1 1  V y : I   f ( x)dx     dx         1  x3 1  x2   1  x2 2(1  x  0    x dx e   x    Phân tích : f ( x)  x2 1  x   1  x2  1  x  1  x  1 x 1  x  1  x   1  x 1  x 3  3      V y : I   f ( x)dx     dx     8    2 x x x x x x                   2    dx f  x 1  x  Hocmai.vn – Ngôi tr ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Hocmai.vn – Website h c tr c n s t i Vi t Nam Khóa h c Luy n thi THPT qu c gia PEN-C: Môn Toán (Th y Nguy n Bá Tu n)  Phân tích : f ( x)  Nguyên hàm – tích phân  A C  x2   A B x  B Ax+B C    x2 1 x x2 1  x x2 1  x  A C   A  1 1 1 x 1   ng nh t h s hai t s :  A B    B   f ( x)      1 x x x x 1 x B  C    4   1  4 V y : I   f ( x)dx        dx    ln x   ln  x   ln  3ln  x x 1 x  x   1 4 1 dx x 1  g      dx  ln  ln  ln  ln 2 x  x  1  x  x  x 2  h   x  11 dx x2  x   x  5  x  11 2x   2  x  5x  x  5x  x  x   x   x  3  Phân tích : f ( x)   1 x 1 2x  1      ln V y : I   f ( x)dx    2  dx  2ln x  x   ln x  5x  x  x   x3 0 x3  x  0 x  dx i   1 x3   x x Phân tích : f ( x)    x2  x  1    x2  x  1     x2  x    x 1 x 1 x 1 x 1 1   1  11 V y : I   f ( x)dx    x2  x    dx   x  x  x  ln x     ln x 1  3  0 x3  x2  x   x2  3x  dx 1 k  Phân tích : f(x)=  Phân tích : x3  x2  x  5x  5x   2x   2x  x  3x  x  3x   x  1 x    A B x  B  A A B 5x      x  1 x   x  x   x  1 x    A B   A  14 19 14 ng nh t h s hai t s :    f ( x)  x   x  x 1  B  A   B  19  V y: 0 19 14   I   f ( x)dx    x    dx   x  19ln x   14ln x   1 32ln  19ln  x  x 1  1 1  3x  3x  l  dx x  x  2  Hocmai.vn – Ngôi tr ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 - Trang | - Hocmai.vn – Website h c tr c n s t i Vi t Nam Khóa h c Luy n thi THPT qu c gia PEN-C: Môn Toán (Th y Nguy n Bá Tu n) Nguyên hàm – tích phân 3x2  3x  3x2  3x  A B C      2 x  3x   x  1  x    x  1  x  1 x   Phân tích : f ( x)    B  C  x2   B  2C  A x  A 2B  C  x  1  x    B  C  A 3      A B  2C    B   f ( x)   x  1  x  1 x  2 A B  C  C    ng nh t h s hai t s ta có : 3   3 V y : I   f ( x)dx       2ln x   ln x     ln  dx      x  1 x    x 1 2 2   x  1 x2 m  dx  x  1   Phân tích : f ( x)  x2  3x  1  A  3x  1 9Cx   3B  6C  x  A B  C  B  3x  1  C  3x    3x  1 ng nh t h s hai t s :  A 9C   2      3B  6C    B    f ( x)  9  3x  1  3x  1  3x  1  A B  C     C   1     V y : I   f ( x)dx     dx   3x  1  3x  1  0   x  1  1 3 1   ln 3x    ln      18  3x  1  3x  1 0 96   x3 dx Bài HKB 2012: I =  x  3x2   t x2 = t x t 2xdx = dt 0 1  x2 x tdt      dx   I=  dt 2 ( x  1)( x  2) (t  1)(t  2)  t  t     1 =  ln t   ln t    ln  ln 2  0 1 Giáo viên: Nguy n Bá Tu n Ngu n Hocmai.vn – Ngôi tr ng chung c a h c trò Vi t T ng đài t v n: 1900 58-58-12 : Hocmai.vn - 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