Toán 12 nguyên hàm tự luận có đáp án

1 F (x) f(x) K f (x)dx kf(x)dx f(x) C k f(x)dx (f(x) g(x))dx x dx dx x ex dx x a dx x f(x)dx C( 1) ln | x | C ex ax ln a g(x)dx C C HDedu - Page cos xdx sin xdx dx cos2 x dx sin2 x sin x C cos x C tan x C cot x C f(u)du f[u(x)] u (x)dx (*) F(x) C dt u (x)dx f(t)dt F(t) C F[u(x)] C K thì: u(x) v (x)dx Hay: udv uv u(x) v(x) u (x) v(x)dx vdu VD1: Tính nguyên hàm: I1 sin x cos xdx = HD t sin x dt cos xdx I1 tdt t2 C sin2 x C 3x 2x dx x3 x VD2: Tính nguyên hàm: I = HD (x3 t x3 x2 I2 x2 5) (3x2 du du u ln|u| C HDedu - Page 3x2 2x 2x)dx ln|x3+x 2+5| C VD3: Tính nguyên hàm sau: I3 xex dx = HD u x e xdx dv Chú ý du ex v xe x I3 dx ex dx xex VD4: Tính nguyên hàm sau: I ex C cos xdx = HD vdu t x, x x cos x dx x cos xdx I4 t t costdt u t dv cos tdt I4 du dt v sin t t sin t sin tdt x sin x 2(t sin t 2cos x cos t C) 2C Tính nguyên hàm sau: I1 2e2x 5dx I3 2cos 2x I5 I7 x (x x(x2 I2 dx I4 2x x dx 2x sin x2 dx I6 sin(3x 1)dx 1)99 dx I8 cos3 x sin xdx 4)5 Tính nguyên hàm sau: I1 x sin 2xdx I2 x cos 2xdx I3 xe2x dx I4 ln xdx HDedu - Page dx P(x) dx, Q(x) Bài toán 1: 1)(x P(x) Q(x) a A x x1 – x2 B x x2 , 0) P(x) Q(x) a A x x0 B , (x x )2 1)(x P(x) Q(x) a A x x1 – x2)(x – x3) thì: B x x2 – x2)2 thì: 1)(x P(x) Q(x) a A x x1 B x x2 1)(x P(x) Q(x) a A x x1 C , x x3 C , (x - x )2 + mx + n) thì: Bx C , x mx n HDedu - Page VD6: Tính nguyên hàm sau: (a) I1 2x dx; x 3x 2x dx x 3x (b) I 2 = HD (a) – Ta có: 2x x 3x 2 2x (x 1)(x 2) A B x A(x 2) B(x 1) (x 1)(x 2) A B 2A B I1 x x (A B)x (2A B) (x 1)(x 2) A B dx x x dx x dx 3ln|x+1| ln|x+2| + C (b) – Ta có: x 2x 3x 2x (x 1)(x 2)2 A(x 2)2 (A B)x2 A B 4A B C 4A 2B C I2 3(x 1) 3(x 2) A B (x 1) 2) B(x 1)(x 2) C(x 1) (x 1)(x 2)2 (4A B C)x 4A (x 1)(x 2)2 A B C 1/3 1/3 3 (x 2)2 dx dx dx dx 3 x x (x 2)2 1 ln | x | ln | x | 3 (x 2) HDedu - Page (x C (x 2)2 C 2B C Tính nguyên hàm sau: 2x dx, x 5x 5x dx, 2x 4x I1 I3 4x dx x 6x 3x dx x 3x I2 2 I4 Tính nguyên hàm sau: 2x dx, x 5x 6x x 2x dx, x x 5x I1 I3 I2 I4 x3 x3 P(x) dx, Q(x) Bài toán 2: P(x) Q(x) x 2x dx 2x 5x x 3x dx 2x 5x P1 (x) , Q(x) R(x) P1 (x) P(x) dx Q(x) P1 (x) dx Q(x) R(x)dx x5 VD7: Tính nguyên hàm sau: I 2x 4x3 x 2x dx = HD x5 x3 I x4 2x 4x3 x 2x 2x 2 x + 2x – 4x3 + cho x2 x3 2x 2x dx x 2x ln x ln|x - 2| C HDedu - Page 2x x 2x Tính nguyên hàm sau: I1 x2 x3 dx, 2x I2 x3 2x 10x dx x 2x Tính nguyên hàm sau: I1 x5 3x3 dx, x2 x I2 HDedu - Page x3 4x dx, x2 2x I3 x3 5x dx 2x e2x I1 I2 +C I5 8(x ln x ln x C I3 I4 cos x I7 C (x2 200 x cos 2x x sin 2x I2 I2 I2 ln x ln x ln x C x2 3ln x x2 ln x 2 x4 I1 I3 2x x2 x3 x3 2x 2x 2x 8ln x 3x 11x x 1 2x e C I4 x ln x x C x I4 ln x 14 ln x I3 11 ln x I4 ln x 5ln x C C C 4x ln x 2x xe ln x C I3 I3 C 12 C x ln x I1 I2 ln x 4ln x I1 I2 cos2x C ln x I1 sin 2x C C 49 ln 2x 16 1)100 C cos4 x C I8 I1 C 4)4 cos(3x-1)+C I6 ) C sin(2x C C ln x ln x C C 2(x 1) C sin2 x cos2 x 1 cos 2x sin2 x cos 2x cos2 x 3sin x sin3x 3cos x cos 3x cos3 x sin3 x Tính nguyên hàm: T1 = (sinx)n dx – T1 (sin x)n dx (sin x)2k dx (sin2 x)k ( cos x) dx (1 cos2 x)k d(cos x) T1 (sin x)n dx (1 t2 )k dt, VD1: Tính nguyên hàm: T 2)k sin2 xdx = HD T sin2 xdx cos 2x dx dx x HDedu - Page cos 2xdx sin 2x C VD2: Tính nguyên hàm: T sin3 3xdx = Chú ý T 3sin3x sin 9x sin3 3xdx sin3xdx cos 3x VD3: Tính nguyên hàm: T sin 9xdx cos 9x 36 C sin5 ( 3x)dx = HD Ta có: T sin5 ( 3x)dx sin5 3xdx sin4 3xd(cos 3x) (1 2cos2 3x cos4 3x)d(cos 3x) (1 cos2 3x)2d(cos 3x) d(cos 3x) cos2 3xd(cos 3x) cos4 3xd(cos 3x) cos 3x cos3 3x cos5 3x C 15 HDedu - Page Tính nguyên hàm: T2 = (cosx)n dx – T2 (cos x)n dx (cos x)2k dx (cos2 x)k (sin x) dx (1 sin2 x)k d(sin x) T2 (cos x)n dx (1 t2 )k dt, VD4: Tính nguyên hàm sau: T 2)k cos4 3xdx = HD T 1 (1 cos 6x)2 dx (1 2cos 6x 4 1 1 cos12x x sin 6x dx 12 1 1 x sin 6x x sin12x C 12 96 1 x sin 6x sin12x C 12 96 cos 3xdx VD5: Tính nguyên hàm: T (sin8 x cos2 6x)dx cos8 x)dx = HD 8x sin8 x cos8 x + cos8 (sin4 x cos x)2 cos 4x cos2 4x 16 2sin4 x cos x sin 2x cos 4x 16 1 cos 4x 1 cos 8x cos 4x 16 16 35 cos 8x cos 4x 64 16 64 HDedu - Page 10 (1 2cos 4x 32 cos2 4x)2 I cos 8xdx 64 sin 8x 512 cos 4xdx 16 sin 4x 64 35x 64 35 dx 64 C T3 = sinf(x).cosg(x)dx [ sin(a b) sin(a b)] cos a cosb [ cos(a b) cos(a b)] sin a sinb [ cos(a b) cos(a b)] sin a cos a VD6: Tính nguyên hàm sau: (a) T1 (b) I cos 3x cos5xdx cos x sin 2x cos 3xdx = HD (a) cos3x.cos5x = T1 (cos 8x (cos 8x cos 2x) cos 2x)dx 1 sin 8x sin 2x (b) Ta có: cos x sin 2x cos 3x T2 cos 6x 24 (sin3x sin x) cos 3x (sin3x cos 3x cos 3x sin x) 1 [ sin 6x (sin 4x sin 2x)] 2 = (sin 6x sin 4x sin 2x) cos 4x 16 HDedu - Page 11 cos 2x C C tann xdx, dx cos2 x dx sin2 x tan x cot x cot nxdx (n dx cos (ax b) dx sin (ax b) C C ) tan(ax b) C a cot(ax b) C a VD7: Tính nguyên hàm: (b) T2 = (1 (a) T1 = tan xdx tan2 x)dx = HD (a) Ta có: T1 = tan xdx sin x dx cos x d(cos x) cos x ln|cosx| C (b) Ta có: T2 = Chú ý dx cos2 x tan2 x)dx tan x C VD8: Tính nguyên hàm: (a) T1 cách làm hoàn toàn (1 tan2 xdx (b) T2 tan3 xdx = HD Ta có: T1 tan2 xdx [(tan2 x 1) 1]dx (tan2 x 1)dx dx tan2 x tan xdx [(tan2 x 1) tan x tan x x C (b) Ta có: T2 (tan2 x 1) tan xdx tan xd(tan x) tan xdx tan xdx tan2 x ln|cosx| C HDedu - Page 12 tan x]dx Tính nguyên hàm sau: T1 (2sin3x T2 T3 T4 cos 2x)dx, T7 sin 2x.coxdx, sin2 (1 2x)dx T8 sin x cos 2x sin3xdx sin4 2xdx, T9 (1 cot2 x)dx, sin3 (2x 1)dx, T5 [3x2 sin5 (1 2x)]dx, T6 [ cos3 3x sin3 ( 2x)]dx T10 cot3 2xdx T11 tan4 2xdx, T12 cot (1 3x)dx Tính nguyên hàm sau: T1 cos 2x(sin4 x cos4 x)dx T2 sin3 x sin3x.dx T3 (sin6 x cos6 x)dx T4 cos x(cos4 x cos3 x)dx Tính nguyên hàm: (a) T1 (sin x cos x)dx sin x cos x (b) T2 cos 2xdx sin x cos x (b) T2 tan5 xdx (b) T2 sin x sin 2x cos5xdx Tính nguyên hàm: (a) T1 sin2 x dx Tính nguyên hàm: (a) T1 sin(x (c) T3 cos2 x cos 2xdx (d) T4 cos3 x cos5xdx (e) T5 sin2 x cos2 2xdx (f) T6 sin x cos x(1 cos x)2 dx )(2 sin 2x)dx HDedu - Page 13 T1 cos3x+2sin2x+C T2 1 x sin(2 - 4x)+C T3 1 x sin2x+ sin4x+C 32 T4 1 cos(2x-1)- cos3 (2x 1)+C T5 x3 1 cos(2x-1)+ cos2 (2x-1) cos5 (2x-1)+C 2 10 T6 1 1 sin3x+ sin3 3x+ cos2x - cos3 2x+C 6 T7 1 cos3x- cos x+C T8 x T9 cot x+C T10 T11 T12 sin 4x cos2 2x tan3 2x sin 6x sin 2x +C ln sin 2x +C tan 2x cot3 (3x 1) x+C cot(3x 1) x+C T1 sin 2x 16 sin 6x+C 48 T2 sin 2x 16 sin 4x 32 T3 x+ sin4x+C 32 T4 sin x sin x 1 x+ sin6x+C 48 sin x 3x sin 2x sin 4x +C 32 (sin x cos x) cos x sin x nên: d(sin x cos x) sin x cos x T1 ln|sinx + cosx| C (b) Vì cos2x = cos2x – sin2x nên T2 T2 (cos x sin x)dx sin x cos x C (a) T1 (x sin x) C tan3 x tan2 xdx T2 tan3 x tan3 xd(tan x) tan4 x T2 cos2 x dx tan x tan2 xdx tan2 x ln cos x C (a) T1 = [ cos x (b) T2 1 sin 6x (c) T3 sin 2x (d) T4 sin 8x 8 (e) T5 x (f) T6 cos 4x x 4 cos 2x sin 4x sin 6x 12 cos4 x C C ] C C C sin 6x sin 2x 2cos3 x sin 3x cos 8x sin 4x 16 sin 2x sin x cos2 x sin x C  ... 3xdx = HD T 1 (1 cos 6x)2 dx (1 2cos 6x 4 1 1 cos12x x sin 6x dx 12 1 1 x sin 6x x sin12x C 12 96 1 x sin 6x sin12x C 12 96 cos 3xdx VD5: Tính nguyên hàm: T (sin8 x cos2 6x)dx cos8 x)dx = HD 8x sin8... C 2B C Tính nguyên hàm sau: 2x dx, x 5x 5x dx, 2x 4x I1 I3 4x dx x 6x 3x dx x 3x I2 2 I4 Tính nguyên hàm sau: 2x dx, x 5x 6x x 2x dx, x x 5x I1 I3 I2 I4 x3 x3 P(x) dx, Q(x) Bài toán 2: P(x) Q(x)... cos 2xdx sin 2x C VD2: Tính nguyên hàm: T sin3 3xdx = Chú ý T 3sin3x sin 9x sin3 3xdx sin3xdx cos 3x VD3: Tính nguyên hàm: T sin 9xdx cos 9x 36 C sin5 ( 3x)dx = HD Ta có: T sin5 ( 3x)dx sin5 3xdx