t tt t t P ì P P A1 = (x + 1)dx(x + 2) sỷ r (x + 1)1(x + 2) = x + + x + t số , t ữ s x+1 1 = x1 lim = lim = =1 (x + 1) (x + 2) x1 x + + x+2 1 = lim = = x2 (x + 1) (x + 2) x2 x + + = lim x = t õ (0 + 1)1(0 + 2) = + + + 12 = + 12 x=1 t õ (1 + 1)1(1 + 2) = + + + 16 = 12 + 31 + = õ t s r số , sỷ r (x + 1)1(x + 2) = x + + x + 2 1 + = =1 = 1 (x + 2) + (x + 1) x ( + ) + + = = (x + 1) (x + 2) (x + 1) (x + 2) (x + 1) (x + 2) số t õ + =0 =1 = + = õ t s r (x + 1)1(x + 2) = x +1 x +1 dx x+1 = ln |x + 1| ln |x + 2| + c = ln +c (x + 1)dx(x + 2) = xdx +1 x+2 x+2 A2 = x (x x2)+(x dx + 5) sỷ r x (x x2)+(x = + + + 5) x x2 x+5 x (x + 2) x+2 0+2 = lim = = x0 x (x 2) (x + 5) x0 (x 2) (x + 5) (0 2) (0 + 5) (x 2) (x + 2) x+2 2+2 = lim = lim = = x2 x (x 2) (x + 5) x2 x (x + 5) (2 + 5) (x + 5) (x + 2) x+2 + = lim = lim = = x5 x (x 2) (x + 5) x5 x (x 2) (5 2) 35 = lim +2 1 x = t õ (11 = + + + = 2) (1 + 5) 1 + 12 x=1 t õ (1 12)+(12 + 5) = + + + + 61 = 12 t tt t t t õ (3 32)+(32 + 5) = + + + 13 + + 81 = 245 t ữủ số , , t ữỡ tr x=3 1 + = 12 1 + = 1 ++ = 24 = = 35 = = + + sỷ r x (x x2)+(x + 5) x x2 x+5 (x 2) (x + 5) + x (x + 5) + x (x 2) x+2 = x (x 2) (x + 5) x (x 2) (x + 5) x ( + + ) + x (3 + 2) 10 x+2 = x (x 2) (x + 5) x (x 2) (x + 5) = + + = số t õ + = = 10 = =3 35 x+2 õ x (x 2) (x + 5) = 5x + (x 2) 35 (x + 5) dx dx dx x (x x2)+(x dx = + + 5) x x 35 x+5 ln |x + 5| + c = ln |x| + ln |x 2| 35 A3 = (3x2 2xx+ 5) (x + 1) dx ợ ữỡ tr õ ax2 + bx + c õ x1, , x2 t ữủ ữợ ax2 + bx + c = a (x x1) (x x2) õ t t 3x2 2x + = (x 1) x+ sỷ r (3x2 2xx+ 5) (x + 1) = x2 (x 1) x + (x + 1) = + x1 x+ + x+1 x2 (x 1) x2 = lim = 5 x1 x1 16 (x 1) x + (x + 1) x+ (x + 1) 3 x2 x + x2 25 = lim = lim = (x 1) (x + 1) 48 x 53 x 53 (x 1) x + (x + 1) = lim = lim x1 x2 (x + 1) x2 = lim = 5 x1 (x 1) x + (x + 1) (x 1) x + 3 t tt t t x = t õ (3x2 2xx+ 5) (x + 1) = x + + + + = x+1 x+ x2 + x = t õ = + =+ + (3x 2x + 5) (x + 1) x1 x+1 33 11 x+ x2 = + + = + + x = t õ (3x 2x + 5) (x + 1) x1 x+1 112 14 x+ + + = = 16 25 tứ õ t õ + + = = 11 33 48 + + = = 14 112 sỷ r (3x2 2xx+ 5) (x + 1) = x2 = + x1 + x+1 5 (x + 1) x+ 3 5 x + (x + 1) + (x 1) (x + 1) + (x 1) x + x 3 = 5 (x 1) x + (x + 1) (x 1) x + (x + 1) 3 5 x2 = x2 ( + + ) + x + + 3 3 ++=1 = 16 25 + = số t õ 3 = 48 5 =0 = 3 õ t2 õ x = (3x 2x + 5) (x + 1) (x 1) x + x2 25 = + 5 16 (x 1) (x + 1) (x 1) x + (x + 1) 48 x + 3 x2 dx 25 dx dx + r (3x2 2x + 5) (x + 1) dx = 16 x 48 x+4 x+ 25 = ln |x 1| ln x + + ln |x + 4| + c 16 48 x1 A4 = (x2 + 4x + 5) (x2 4) dx ú ỵ r ữỡ tr ax2 + bx + c = ợ = b2 4ac < t t t ax2 + bx + c = a (x x1 ) (x x2 ) tr õ x1 = + i, x2 = i, i2 = õ t õ x2 + 4x + = (x + + i) (x + i) sỷ r x1 x1 = = + + + 2 + 4x + 5) (x 4) (x + + i) (x + i) (x 4) x+2+i x+2i x2 x+2 (x 1) (x + + i) x1 13 = lim = lim = i 2 x2i (x + + i) (x + i) (x 4) x2i (x + i) (x 4) 34 34 (x2 t tt t t (x 1) (x + i) x1 13 = lim = + i x2+i (x + + i) (x + i) (x2 4) x2+i (x + + i) (x2 4) 34 34 (x 1) (x 2) x1 = lim = lim = x2 (x + 4x + 5) (x 2) (x + 2) x2 (x + 4x + 5) (x + 2) 68 x1 (x 1) (x + 2) = lim = = lim 2 x2 (x + 4x + 5) (x 2) x2 (x + 4x + 5) (x 2) (x + 2) 13 13 1 i + i x1 õ t õ (x2 + 4x + 5) (x2 4) = x34+ +34i + x34+ 34i + x 68 + x+2 13x 27 = + + 17 (x + 4x + 5) 68 (x 2) (x + 2) = lim + + + sỷ r (x2 + 4xx+5)1 (x2 4) = x2x + 4x + x x + x = t õ 51 21 + 12 = 201 1 x = t õ + + =0 10 10 3 1 x = t õ + ++ = 26 26 65 1 1 x = t õ + + + = 37 37 148 tứ õ t õ ữủ 1 1 + = 2 20 + + = 10 10 1 + ++ = 26 26 65 1 1 37 = ++ 10 = 10 30 30 27 + = 26 650 650 22 11 13 = 17 = 27 + + = + = 148 37 555 1110 x1 13x 27 = + + 2 (x + 4x + 5) (x 4) 17 (x + 4x + 5) 68 (x 2) (x + 2) + 37 17 = 68 = õ + + + sỷ (x2 + 4xx+5)1 (x2 4) = x2x + 4x + x x + x = (x + ) x2 + x2 + 4x + (x + 2) + x2 + 4x + (x 2) x = x3 ( + + ) + x2( + + 2) + x (4 + 13 3) + (4 + 10 10) = ++ =0 số t õ + + ( ) = + + = + 13 = + 13 ( ) = 10 + 10 = 10 + 10 ( ) = 13 = 17 = 27 + + = + 16 = 10 + 20 = = 17 = 68 = t tt t t 13x 27 õ (x2 + 4xx+5)1 (x2 4) = 17 (x + + + 4x + 5) 68 (x 2) (x + 2) 54 2x + x1 13 dx (x2 + 4x + 5) (x2 4) dx = 34 x2 + 4x13+ dx + 681 x dx + x+2 (2x + 4) dx dx dx 13 dx + + = 2 34 x + 4x + 17 x + 4x + 68 x2 x+2 d x + 4x + 13 1 dx dx dx = + + 2 34 x + 4x + 17 x2 x+2 (x + 2) + 68 1 13 arctan (x + 2) + ln |x 2| + ln |x + 2| + c = ln x2 + 4x + 34 17 68 x A5 = x3 + dx + sỷ r x3 x+ = (x + 1) (xx2 x + 1) = x + + 3 x i x + i 2 2 x = = lim x1 x x + x = = lim i 6 x 21 + 23 i i (x + 1) x + 2 x = lim = + i 6 x 12 23 i (x + 1) x i 2 3 i + i x õ x3 + = (x + 1) + 6 + 6 = (x1+ 1) + (x2x+x1+ 1) 1 3 x i x + i 2 2 sỷ x3 x+ = x + + x2x +x + x = t õ = + 1 x = t õ = + + 2 2 x = t õ = + + 3 +=0 = 1 ++= = 2 2 + + = = 3 x x+1 = + x3 + (x + 1) (x2 x + 1) õ t õ ụ sỷ r x3 x+ = x + + x2x +x + 1 3 x = x2 x + + (x + ) (x + 1) t tt t t + =0 số t õ + + = +=0 x3 x dx = +1 = = dx + x+1 dx + x+1 dx + x+1 = = = 2x + dx x+1 2x + 1 dx + x x+1 d x x+1 + x2 x + x2 dx x+1 dx x + 1 2x = ln |x + 1| + ln x x + + arctan +c 3 A6 = x4dx+ sỷ r x4 + = x2 + px + x2 px + = x4 + p2 x + ỗ t t õ p2 =0 p = õ t t x4 + = x2 + 2x + x2 2x + 1 = x + 2x + x2 2x + + + + = 2 2 2 2 + i x+ i x + i x i x+ 2 2 2 2 số , , , ữ s x2 = lim x i 2 i x+ 2 = lim 2 x + i 2 2 x+ + i 2 x2 2x + x2 = lim 2 x i 2 2 x i 2 = lim x + i 2 x4 1+ = x2 + 2x + = + 2i 8 = 2i 8 2x + = + 2i 8 = 2i 8 2 x + i x2 + 2x + 2 2 2 2 + i x+ i + i 8 2 8 x2 + 2x + 2 x+ + i 2 + t tt t t + 2 + i 8 2 x i 2 2 i 8 + x2 2x + 2 x + i 2 1 2 x+ x+ + = 2 x + 2x + x 2x + sỷ r x4 1+ = x+ + x+ x + 2x + x 2x + 1 x = t õ = + + 2 2 2+ 2+ x = t õ + = 1 1 1 + + x = t õ = 2+ 2+ 2 2 1 + + + x = t õ = 5 + =1 1 1 + + = tứ õ t õ 21 21 21+ 21+ + + = 2 + 2 + 2 2 1 + + + = + =1 + + = + ++ 3+2 + 3+2 = + + + = + =1 + = = == + a 2+2 + 3+2 = = 2 + = 4 2 x+ x+ + = 2 x +1 x + 2x + x 2x + t õ ữủ ụ sỷ r x4 1+ = x + x + + + 2x + x 2x + = (x + ) x2 2x + + (x + ) x2 + 2x + = ( + ) x3 + + + + x2 + + + x + + x2 số t õ ữủ s = +=0 +=0 + + + = + = 1 == + + = 2 + = 2 = + =1 =1 t tt t t 1 2 x+ x+ õ x4 + = + x + 2x + x 2x + 1 1 x+ x+ t õ x4dx+ = 22 2 dx + 2 2 dx x + 2x + x 2x + 1 x+ 1 2x + dx 2 dx = dx + õ x + 2x + x2 + 2x + x2 + 2x + d x + 2x + 1 dx = + 4 x2 + 2x + 1 x+ + 2 1 = ln x2 + 2x + + arctan 2x + + c1 2 1 x+ 2x + dx 2 dx = dx + õ 2 x 2x + x 2x + x 2x + 2 d x 2x + dx 1 = + 4 x2 + 2x + 1 x + 2 1 = ln x2 2x + + arctan 2x + c2 2 dx x2 + 2x + 1 x4 + = ln + arctan 2x + + arctan 2x + c x 2x + 2 2 dx A7 = x8 + sỷ r x8 + = x4 + px2 + x4 px2 + = x8 + p2 x4 + ỗ t t ữủ p2 = p = x8 + = x4 + 2x2 + x4 2x2 + õ x4 + 2x2 + = x2 + qx + x2 qx + = x4 + q2 x2 + ỗ t t õ q2 = q = x4 + 2x2 + = x2 + 2x + x2 2x + õ x4 2x2 + = x2 + rx + x2 rx + = x4 + r2 x2 + ỗ t t õ ữủ r2 = r = + x4 2x2 + = x2 + + 2x + x2 + 2x + õ t õ 2x + x2 2x + x2 + + 2x + x2 + 2x + 1 2x + 2x + 2 + 2x + = + + + x +1 x2 + 2x + x2 2x + x2 + + 2x + + 2x + + x2 + 2x + x8 +1 = x2 + t tt t t ú ỵ dx = ax2 + bx + c a dx , 4ac b2 > b c x2 + x + a a dx dx = (1) = 2 2 a a (x p) + q b 4ac b + x 2a 2a 4ac b2 t p = b ,q = , x = p + qt 2a 2a õ = aq1 t2 1+ dt = 2 t2 1+ dt = 2 arctan t 4ac b 4ac b 2ax + b = arctan +C 4ac b2 4ac b2 r õ t t t = x q p = 2ax + b 4ac b xdx 2ax + b b 2ax + b 1 b õ ax2 + bx + c = 2a ax dx = dx dx + bx + c 2 2a ax + bx + c 2a ax + bx + c b dx = ln ax2 + bx + c +C 2a 2a ax2 + bx + c ợ a = c = 1, = b2 > t õ xdx dx Ax + B dx = A +B 2 x + bx + x + bx + x + bx + 2x + b A 2B Ab arctan + ln x2 + bx + + C = 2 4b b +2 õ A7.1 = 81 2 2x dx = x + 2x + 2+ 2x + 2 + = arctan ln x2 + 2x + + C1 16 2+ 2x + dx = A7.2 = x2 2x + 2+ 2x 2 2 = arctan ln x 2x + + C2 16 2+ + 2x + A7.3 = dx = x2 + + 2x + 2 2x + 2+ = arctan + ln x2 + + 2x + + C3 16 2 + 2x + A7.4 = dx = x2 + 2x + 2 2x 2+ = arctan ln x2 + 2x + + C4 16 2 õ t õ A7 = A7.1 + A7.2 + A7.3 + A7.4 õ x8 + = x8 = = cos ( + k2) + i sin ( + k2) t tt t t + k2 + k2 + i sin ợ k = 0, , 8 õ sin = sin = cos = = sin = cos = 2, sin 2+ , 8 8 = cos = 2, cos 8 dx (2k + 1) (2k + 1) õ t õ + x8 = ln x2 2x cos +1 ìcos 8 x = cos k=0 + (2k + 1) x sin (2k + 1) ì sin + + C, k = 0, , arctan (2k + 1) k=0 x cos tr t t số , , , , số t tự ỳ t số ổ t ũ t õ t tr t ró t s q t ợ t tữỡ tỹ ữ s I1 = (x x3)+(x27+ 3) dx sỷ r (x x3)+(x27+ 3) = x + x + x + 27 = (x + 3) + (x 3) = x ( + ) + (3 3) số t õ + =1 =5 = =9 (x x3)+(x27+ 3) dx = x dx x +4 dx = ln |x 3| ln |x + 3| + c I2 = (x 8x3)(x17+ 4) dx sỷ r (x 8x3)(x17+ 4) = x + x + = (x(x+4)3)+(x +(x4) 3) = x ((x+)3)+(x(4+ 4) 3) tứ õ t õ + =8 =1 = 17 =7 (x 8x3)(x17+ 4) dx = x dx + x +7 dx = ln |x 3| + ln |x + 4| + c 26 I3 = x2 7x dx 6x 16 26 7x 26 ( + ) x + (2 8) sỷ r x2 7x = = + = 6x 16 (x 8) (x + 2) x8 x+2 (x 8) (x + 2) số t õ + =7 =3 = 26 =4 26 x2 7x dx = dx + dx = ln |x 8| + ln |x + 2| + c 6x 16 x8 x+2 150 I4 = 7x x+3 75x dx 25x 150 7x2 + 75x 150 sỷ r 7x x+3 75x = = + + 25x x (x 5) (x + 5) x x5 x+5 t tt t t x2 + õ lim x 6i (x + 2) õ t õ + i = 10 6 Im + i 10 = = = 6 Im + i 10 = Rec = + i + 10 10 = lim x2 + 4+1 = = = x +6 4+6 10 = lim x2 + x2 + x2 x2 t õ ữủ 10x = lim x2 x2 + (x2 + 6) = 20 = 100 1 2x + + 10 (x + 6) (x + 2) 2(x + 2)2 (x2 + 6) (x + 2) I10 = (x2 x x++1)1(x2 + 1) dx + sỷ r (x2 x x++1)1(x2 + 1) = x2x +x + + x x2 + ữỡ tr x2 x + = x = 21 23 i x+1 3 õ lim = i 2 x + i x + = õ t õ 2 3 i Im 2 = = 3 = Rec i + |1| = + = 2 2 2 õ ữỡ tr x2 + = x = i x+1 õ xi lim = + i x x+1 õ s r = Im [1 + i] = = Rec [1 + i] = +2 x1 + õ t õ (x2 x x++1)1(x2 + 1) = x2x x+1 x +1 (x2 x+1 dx = x + 1) (x2 + 1) t tt t t 2x dx + x2 x + d x2 x + + x x+1 dx 2x dx + dx x2 x + x2 + x2 + d x +1 dx dx + 2 2 x +1 x +1 x + 2x 1 + ln x2 + arctan x + c = ln x2 x + + arctan 2 x +1 2x 1 = ln + arctan arctan x + c 2 x x+1 = = t tỹ I1 = I2 = I3 = x dx (x 1) (x + 1) (x + 2) x2 dx x2 3x + (x2 1) dx 2 (x + x + 2) (x2 + 1) x2 dx (x3 + 2) (x 1)3 I4 = Pì PP ìẹ P t số số f x, a2 x2 dx f x, x2 a2 dx f x, x2 + a2 dx t ; 2 a x= t 0; ; cos t 2 x = a tan t t 0; a+x dx x, x = a cos 2t t 0; ax (x a) (b x) dx x = a + (b a) sin2 t t 0; f f x, x = a sin t tõ tt ổ tự ữỡ số f x, a2 x2 dx A1 = 1xdx x2 t x = sin t dx = cos tdt xdx = sin t cos tdt = sin t cos tdt |cos t| 1x sin t sin t cos tdt = sin tdt = cos t + c, t < , cos t = sin t cos tdt = sin tdt = cos t + c, t < , cos t t tt t t < x < 1, xdx = xdx + x x 1x (1 x) (1 + x) t x + + x = t x = sin t, + x = cos t õ t õ x = 2sin2t dx = sin2t cos tdt 2sin t sin t cos tdt xdx = cos 2tdt = sin 2t + c = sin t cos t + x x õ xdx = d x2 1 x2 d x2 = x2 + c = x2 x2 dx A2 = 2+x+ 2x ú ỵ + x+ x = 4, õ t t + x = sin t, x = cos t, t ; + x = 4sin2 t dx = sin t cos tdt dx sin t cos tdt (sin t + cos t)2 =4 =2 dt sin t + cos t sin t + cos t 2+x+ 2x dt dt = (sin t + cos t)dt = (sin t + cos t)dt sin t + cos t sin t + dt = (sin t + cos t)dt t t sin + cos + 8 t = (sin t cos t) ln tan + +c xdx = t A3 = x2 x2 dx t x = sin t dx = cos tdt ợ x = t = 0, x = A3 = sin2 t = (1 + cos 2t)2 dt = + cos 2t + cos2 2t dt + cos 2t + 1 + cos 4t dt = 1 = 3t + sin 2t + sin 4t cos4 tdt = t= sin2 t cos2 tdt = sin2 t cos tdt = (3 + cos 2t + cos 4t) dt = + 64 t tt t t ợ x 23 t õ x2 = x + x ú ỵ r x+1+x=2 õ t t x = sint, + x = cos t, dx = sin t cos tdt x = t = , x = 23 t = 12 12 A3 = 32 12 sin2 t.cos2 t sin t cos t sin t cos tdt = 32 12 12 = A4 = = tan t = 12 = + 64 cos tdt = 2 2sin t cos t = 2sin t A4 = dt cos2 t ợ x t õ x2 = 2x 2+x t x = 2 sin t, + x = 2 cos t, õ x = 2 sin t, dx = sin t cos tdt x = t = , x = t = , + cos 8t dt cos 4t + x = t = 0, x = t = , 2 2sin t dt = dx (2 x2 ) x2 cos tdt 2 x = sin t dx = cos tdt 12 1 t sin 4t + sin 8t 2 16 t A4 = t cos 4t sin4 2tdt = = sin4 t.cos4 tdt sin t cos tdt = 8sin t.cos2 t.2 sin t cos t 1 = tan t tan t t A5 = = x 2+x dt = sin t.cos2 t 1+ = d (tan t) tan2 t 1 0= 2 dx 1+ x (x + 1) 12 õ x (x + 1) = 1 x2 + x = x2 + .x + 4 = 1 x+ 2 t tt t t t x + 12 = 21 sin t dx = 12 cos tdt x = 21 t = 0, x = t = , A5 = 12 = = 2 cos tdt cos tdt dt dx = = = + cos t 2 + cos t 1 1 sin t 1+ 0 1+ x+ 4 t t 2 d tan tan dt = arctan 2 = t t 2 3 + 2cos2 tan2 + 0 2 94 + arctan arctan = = 3 18 3 3 ú ỵ x + x + = t x = sin t, + x = cos t x = sin2 t dx = sin t cos tdt x = t = , x = t = 0 dx 1+ x (x + 1) 21 sin t cos tdt = + sin t cos t = = 2 (2 + sin t cos t 2) dt + sin t cos t dt = + sin t cos t d (tan t) tan2 t + tan t + = 2 t u = tan t, ợ d (tan t) = 2J 2 tan t + + t = u = 0, t = u = d (tan t) dt J = = 1 3 tan t + t+ + + 0 4 3 t t + = tan v dt = + tan2v dv t = v = , t = v = , 3 + tan2 v dv dt 2 õ = = 3 3 tan2 v + t+ + 4 94 t õ ữủ I = 2J = = 18 3 dv = v = 3 f x, x2 a2 dx B1 = x x2 dx t tt t t t x = cos1 t , t = = ; x2 x2 dx dx = t B2 = t sin t dt cos2 t sin t sin2 t sin t cos t sin t cos2 t cos2 t sin t dt = dt = dt cos t cos t cos t cos t cos3 t sin2 t cos2 t cos2 t dt = dt = d (tan t) cos4 t cos4 t cos2 t tan3 t + tan2 t d (tan t) = +c B1 = 0; 2 sin t dx = dt x= cos t cos2 t 4 sin t cos2 t B2 = dt = 16 cos t cos2 t ợ tr x = t = 0, x = t = = 16 sin2 t d (sin t) = 16 cos6 t sin2 t sin2 t sin2 t P t = lim x1 = lim x1 = lim x1 r u2 (1 u2 ) = 1, = lim (1 + u)3 x1 (2) 16u2 d (sin t) 3 16u2 , (1 + u)3 16u2 = 1, = lim = 1, = lim x1 (u 1)3 (u 1)3 16u2 =2 (u 1)3 (1 u) (1 + u) u2 (2) 16u2 16u2 16 = sin2 t dt cos5 t (u 1)3 (1 + u)3 16u2 = + + + + + 3 u (u 1) + u (1 + u) (1 u) (1 + u) (u 1) (1 + u)3 sin2 t = t u = sin t, t = u = 0, t = u = õ sin2 t cos2 t sin t dt = 16 cos2 t cos2 t sin2 t sin2 t 3 = x1 16u2 (1 + u)3 = , = 1, 1 1 + u (u 1) (1 + u) (1 + u) (u 1) (1 + u)3 d (sin t) = 1 1 + du u (u 1) (1 + u) (1 + u) (u 1) (1 + u)3 = t tt t t = 16 u2 (u 1)3 (1 + u)3 du = u3 + u (u2 1) u1 + ln u+1 = 28 3+ln 0 + t) t t = x2 t2 = x2 tdt = xdx dxt = d (x x+t t t õ t2 = x2 (t x) (t + x) = t u = t + x t x = u4 x = t = 2, x = t = + 16 u3 4 du 2 x x 4dx = 16 u + u u u u = + u5 16 4 u4 ln |u| + c + u4 64 u4 B2 = + ln |u| u 64 2+ du = t õ 4+2 = 28 ln + 2 t B3 = dx (x2 1) x2 tdt ợ tr x= t x = cos1 t dx = sin cos t B3 = sin tdt = sin t 62 = 1 cos2 t ,x = t = cos tdt = sin2 t = 1 cos2 t cos2 t t= d (sin t) sin2 t t t = x2 t2 = x2 tdt = xdx, dtx = dxt dt d (x + t) õ t s r dxt = dxx + = +t x+t t t õ t2 = x2 (t + x) (t x) = t u = t + x t x = u1 ợ tr x = u = 3, x = u = + 2+ B3 = = u2 2+ du u u 2+ udu =4 2 u (u2 1) 62 = t B4 = 2+ =2 d u2 (u2 1) dx (x + 1) 2x x2 12 t tt t t 0 õ B4 = dx 12 (x + 1) x+1 d = (x + 1)2 21 (x + 1) 2 ds = s2 ú ỵ t t t t s r số ữủ arc õ t ữ s + t) t t = s2 41 tdt = sds dst = d (s s+t t= cos s t t õ (t s) (t + s) = 1, t u = t + s t s = u1 s = u = + , s = u = + 215 ds du = ln |u| + c õ t õ = 2u 2 s2 41 t ữủ B4 = ln |u| 2+ 15 1+ + 15 2+ = ln t B5 = dx (x + 1) + 2x x2 12 dx = (x + 1) + 2x x2 12 dx (x + 1) (3 x) (x + 1) 21 dx = 12 (x + 1)2 3x x+1 3x tdt dx = x+1 (x + 1)2 x = t = 7, x = t = dx tdt = = 2t 2 3x (x + 1) x+1 t t = 21 õ dx = (x + 1) + 2x x2 21 dx 12 (x + 1) (x 1)2 sỷ r i2 = t t = (x 1)2 t2 i2(x 1)2 = (t ix + i) (t + ix i) = dt d (ix + t i) õ tdt = i (x 1) d (ix) dix = = t i (x 1) t + ix i t u = t + ix i õ t õ t ix + i = u4 t õ t = 21 u + u4 , x = 2i1 u u4 + t tt t t r d (ix) d (ix) = i (x + 1) t = du =2 u + 2i u u du u2 + 4ui (x 1)2 du d (u + 2i) =2 +c =2 = 2 u + 4ui + 4i u + 2i (u + 2i) ợ tr x = u = 27 32 i; x = u = i 3i 7 t õ ữủ B5 = u + 2i = + i + i = i (x + 1) f x, 2i x2 + a2 dx t C1 = x2 + dx t x = tan t dx = + tan2t dt x = t = 0, x = t = õ 0 d (sin t) = sin t = sin t + + sin t (1 sin t) (1 + sin t) 4 d (sin t) cos4 t d (sin t) = dt = cos3 t tan2 t + 1 + tan2 t dt = x2 + 1dx = 1 + + sin t sin t d (sin t) = 1 + sin t d (sin t)+ 1 sin t d (sin t)+ d (sin t) (1 + sin t) (1 sin t) 1 1 + sin t + sin t + + d (sin t) (1 + sin t) (1 sin t) (1 + sin t) (1 sin t) 1 + sin t = + ln + ln + = sin t 2 = t t = x2 + t2 = x2 + tdt = xdx + tdx tdt + tdx d (x + t) õ dxt = (x(x++t)t)dxt = xdx = = (x + t) t (x + t) t x+t 2 t t õ t = x + (t + x) (t x) = t u = t + x t x = u1 x = t = 1, x = t = + t tt t t õ x2 x2 + 1dx = + dx = u+ x2 + 1 = + ln |u| + c 8u ữủ u8 + 12 ln |u| 8u1 u u du = u+ + du u u u2 t õ t C2 = 1+ = + ln + 2 ln x2 + x2 dx t x = tan t dx = + tan t dt x = t = , x = x = 3 ln x2 ln tan2 t + x2 dx = 1 + tan2 t + tan2 t dt cot2 x + ln (cos t) dt = = ln (cos t) d (cot t) = cot t ln (cos t) + sin t cot t dt = ln ln cos t 2 + 12 t t = x2 + t2 = x2 + 1, tdt = xdx + tdx d (x + t) õ t õ dxt = (x(x++t)t)dxt = xdx = (x + t) t x+t 2 t t õ t = x + (t + x) (t x) = t u = t + x t x = u1 x = t = + 2, x = t = + 1 u x12 ln + x2dx = ln u+ du 2 u u u u 1 ln u+ 1 u u du + c u+ d = +2 = ln 1 u u u u2 u u u 1 1 ln u+ ln u+ du u u = +2 + c = + arctan u + c 1 u +1 u u u u u+ t ữủ t tt t t 1 2+ ln u+ u C2 = + arctan u u 1+ u ln = + ln + arctan + arctan + 3 ln ln = + 2 12 t C3 = x2 + dx x x4 + 1 x2 + 1 dx = x x4 + 1 x2 + d x2 x2 x4 + 1 = + tan2 t dt t = tan t d x = t = , x = t = x2 x2 C3 = (tan x + 1) d (tan x) tan x = tan2 x + = (tan x + 1) cos2 x + tan2 x dx sin x (tan x + 1) dx = sin x d (sin x) + sin2 x 1 + dx = cos x sin x d (cos x) cos2 x sin x cos x ln + ln sin x + cos x + = ln + ln 2 ln ln 2 = ln ln = t C4 = x3 x2 + dx t x = tan t dx = + tan2t dt x = t = 0, x = t = C4 = tan t tan2 t + sin t dt = cos6 t + tan t dt = 1 1 = d (cos t) = + cos4 t cos6 t 3cos3 t 5cos5 t 2 2 2+2 = + + = 15 15 t = x2 + t2 = x2 + 1, tdt = xdx cos2 t d (cos t) cos6 t t tt t t + tdx tdt + tdx d (x + t) õ dxt = (x(x++t)t)dxt = xdx = = (x + t) t (x + t) t x+t 2 t t õ t = x + (t + x) (t x) = t u = t + x t x = u1 ợ tr x = u = 1, x = u = + du u x3 x2 + 1dx = 321 u+ u u u t õ 1 u+ C4 = 160 u u u u u 1 u+ 24 u u u+ u du u2 2 d u+ u 2 1 u+ u+ d u+ u u u 1 1 = u+ u+ +c 160 u 24 u 32 = 32 = 32 = u+ 1+ 2 = 160 2 24 2+2 + = 15 15 t C5 = dx cos x sin 2x + P t sin 2x + = sin x cos x + = cos2x tan x + cos2 x = 2cos2 x tan2 x + tan x + = 2cos2 x tan x + 2 + õ dx dx = cos x sin 2x + 2cos2 x tan x + 12 + 34 0 t tan x + = tan t + tan2x dx = 23 + tan2t dt x = t = , x = t = 3 2 + tan t dt + tan t dt dt C5 = = = = cos t 2 3 tan2 t + 2 tan t + 6 4 sin t 1 = ln = ln ln sin t + 2 2 t C6 = t x = tan t dx = d (sin t) sin2 t 6 3 dx (x2 + 9) + tan2 t dt ợ x = t = 0, x = 3t= t tt t t C6 = + tan2 t + tan2 t dt = + tan2 t cos tdt = sin t = 18 t t = x2 + t2 = x2 + 9, tdt = xdx + tdx dx + dt d (x + t) = = õ dxt = (x(x++t)t)dxt = xdx (x + t) t x+t x+t 2 t t õ t = x + (t + x) (t x) = t u = t + x t x = u9 x = t = 3, x = 3 u = + 3 d u2 + dx du =4 =2 +c = 2 2 u +9 (x2 + 9) t õ f x, u+ u C6 = u +9 a+x ax u 6+3 3 (u + 9) + 3 + = = 36 18 18 dx t D1 = x 1+x dx 1x t x = cos 2t dx = sin 2tdt x = t = , x = 12 t = D1 = cos 2t sin 2t + cos 2t dt = cos 2t cos 2t sin 2t = cos 2t sin t cos t cos t dt = sin t cos 2t.cos2 tdt = cos 2t (1 + cos 2t) dt + cos 4t cos 2t + dt = sin 2t sin 4t t = 2cos2 t dt 2sin2 t + = + 12 t + x + x = õ t + x = sin t, x = cos t r + x = 2sin2t dx = sin t cos tdt x = t = , x = 12 t = D1 = 2sin2 t sin t sin t cos tdt =4 cos t 2sin2 t sin2 tdt t tt t t + cos 4t cos 2t dt cos 2t (cos 2t 1) dt = =2 + = + 12 = sin 4t sin 2t + t t D2 = xdx 3+x+ 3x x 3 3+x = xdx dx 3+x+ 3x 3x 1+ 3+x t x = cos 2t dx = sin 2tdt x = t = , x = t = cos 2t cos 2t 4 + cos 2t 6cos2 t sin 2tdt sin 2tdt = õ D2 = cos 2t 6sin2 t 1+ 1+ + cos 2t 6cos2 t =6 cos 2t sin t cos tdt = 6 sin t + cos t (cos t sin t) sin t cos tdt =6 sin tcos2 tdt6 sin2 t cos tdt = 6 = 6 cos3 t 3 + sin t cos2 td (cos t)6 sin2 td (sin t) = + t + x + x = õ t + x = sin t, x = cos t út x t õ ữủ x = 6sin2t = 2sin2t , dx = 12 sin t cos tdt x = t = , x = t = D2 = = 2 36 2sin t sin t cos tdt = (sin t + cos t) 36 (sin t cos t) (cos t + sin t) sin t cos tdt (sin t + cos t) 6 (sin t cos t) sin t cos tdt = 6 sin2 t cos tdt 6 =6 sin tcos2 tdt sin2 td (sin t) + 6 cos2 td (cos t) t tt t t 6 sin3 t + cos3 t = f x, =2 62 (x a) (b x) dx t E1 = (x 1) (2 x) dx t x = + sin t dx = sin t cos tdt x = t = 0, x = t = 2 õ E1 = + sin2 t sin t cos t sin2 t dt sin t cos t =2 0 = sin2 t.cos2 tdt sin2 t sin2 t dt = 1 (1 cos 4t)dt = t sin 4t sin2 2tdt = 0 = ú ỵ x + x = õ t t x = sin t, x = cos t õ x = sin2t dx = sin t cos tdt x = t = 0, x = t = E1 = sin2 t.cos2 t = sin2 2tdt = 1 (1 cos 4t)dt = t sin 4t =