TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN NGUYấN HM V PHNG PHP TèM NGUYấN HM Khỏi nim nguyờn hm v tớnh cht Khỏi nim nguyờn hm f (x) Cho ham sụ xac inh trờn f (x) cua ham sụ trờn K K F (x) Ham sụ F Â(x) = f (x), " x ẻ K nờu: F (x) c goi la nguyờn hm f (x) Nờu la mụt nguyờn ham cua f (x) K ham sụ trờn la: ũ f (x) ìdx = F (x) +C , const = C ẻ trờn K thi ho nguyờn hm cua Ă f (x), g(x) Tớnh cht: Nờu luụn cú: la ham sụ liờn tc trờn ã ũ f Â(x)dx = f (x) +C ã ũ ộởờf (x) g(x)ựỷỳdx = ũ f (x)dx ũ g(x)dx ã K va kạ thi ta ũkf (x)dx = kũ f (x)dx Bng nguyờn hm ca mt s hm thng gp (vi C l hng s y) a ũ x ìdx = xa +1 +C a +1 ũ x ìdx = ln x +C ũx ìdx = ũ sin x ìdx = - cosx + C ũ cosx ìdx = sin x +C ũ sin +C x x ìdx = - cot x + C n ũ(ax + b) ìdx = 1 (ax + b)n+1 ì +C a n +1 ũ ax + b ìdx = a ìln ax + b + C ũ (ax + b) 1 ì +C a ax + b ) =ũ sin(ax + bdx cos(ax + b) + C a ìdx = ũ cos(ax + b) ìdx = a ìsin(ax + b) + C ũ sin (ax + b)dx = cot(ax + b) + C a http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 1| TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN 1 ũ cos (ax + b)dx = a tan(ax + b) +C ũe ũe 2 x ũ cos x ìdx = tan x +C ìdx = ex + C x ũa ìdx = ax+b ax +C lna Nhn xột Khi thay x ũx ìdx = ax+b ìe +C a dx x- a = ìln +C 2a x +a - a (ax + b) bng thi lõy nguyờn ham nhõn kờt qua thờm ì a Mt s lu y Cn nm vng bang nguyờn ham Nguyờn ham cua mụt tich (thng) cua nhiờu ham ham sụ khụng bao gi bng tich (thng) cua cac nguyờn ham cua nhng ham phn Muụn tim nguyờn ham cua mụt ham sụ, ta phai biờn ụi ham sụ mụt tụng hoc hiờu cua nhng ham sụ tim c nguyờn ham (da vao bang nguyờn ham) Dng toỏn TNH NGUYấN HM BNG BNG NGUYấN HM A PHNG PHP GII TON Tich cua a thc hoc ly tha Tich cac ham m Cha cn PP ắắ ắ đ PP ắắ ắ đ PP ắắ ắ đ khai trin khai trin theo cụng thc m chuyn vờ ly tha Tich lng giac bc mụt cua sin va cosin PP ắắ ắ đ khai trin theo cụng thc tich tng ã ã | THBTN sinax.cosbx = 1ộ ự ờsin(a + b)x + sin(a - b)xỳ ỷ 2ở sinax.sinbx = 1ộ ự ờcos(a - b)x - cos(a + b)xỷ ỳ 2ở TON HC BC TRUNG NAM TI LIU HC TP TON 12 ã cosax.cosbx = Chuyờn : NGUYấN HM TCH PHN 1ộ ự ờcos(a + b)x + cos(a - b)xỳ ỷ 2ở Bc chn cua sin va cosin PP ắắ ắ đ H bc B - BI TP VN DNG BT Tim nguyờn ham cua cac ham sụ sau (gia s iờu kiờn c xac inh): Phng phỏp: Da vo bng nguyờn hm ca cỏc hm s v dng cỏc tớnh cht nguyờn hm a) a f (a) = 3a2 + ì F (a) = a3 + S: a2 +C b) f (b) = 2b3 - 5b + S: b4 5b2 F (b) = + 7b + C 2 c) c3 F (x) = c - 3c + - 8c + C f (c) = 6c5 - 12c3 + c2 - S: F (x) = d) f (x) = (x - 3x) ì(x + 1) ì S: x4 2x3 3x2 +C F (x) = - e) f (x) = (3 - x) S: (3 - x)4 +C f (x) = f) 1 - x2 - ì x F (x) = S: x3 x +C x 3 http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 3| TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN F (x) = 2x g) f (x) = 10 S: 102x +C 2ln10 f (x) = x3 - 4x + h) ì x F (x) = S: x4 + 2x2 + 3.ln x + C f (t) = i) 2t + ì t2 F (t) = S: ìt - + C t f (x) = j) x- ì x2 F (x) = ln x + S: +C x f (x) = 2sin2 k) x ì F (x) = x + sin x + C S: l) f (x) = cos x S: 1 F (x) = x + sin2x + C m) f (x) = tan2 x F (x) = tan x - x + C S: f (x) = n) ì sin2 x.cos2 x F (x) = S: +C sin2 2x | THBTN TON HC BC TRUNG NAM TI LIU HC TP TON 12 o) Chuyờn : NGUYấN HM TCH PHN F (x) = - f (x) = 2sin3x cos2x S: cos5x - cosx + C x p) x f (x) = e (e - 1) S: F (x) = e2x - ex +C ổ e- x ữ ữ f (x) = ex ìỗ + ì ỗ ữ ỗ ỗ cos2 x ữ ố ứ F (x) = 2ex + tan x + C q) S: I = ũ( x + x) ìdx r) S: 23 I = ìx + C s) ổ ữ ữ ỗ I = ũỗ x + ữìdx ì ỗ 2ữ ỗ ố x ứ S: I = x3 + 33 x + C I = x x +C I =ũ t) x + 3 x + 5 x ìdx ì F (x) = x + S: 25 x + x +C u) I = ũ 4sin2 x ìdx S: I = 2x - sin2x + C I =ũ v) + cos4x ìdx I = S: x sin4x + +C http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 5| TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN I = 3sin x - I = ũ(3cosx - ) ìdx ì x- w) S: 3x- +C ln3 x) I = ũ (tan x - 2cot x)2.dx I = tan x - 4cot x - 9x + C S: y) I = I = ũ u.(u - 4).du S: 33 u - 33 u4 + C BT Chng minh F (x) f (x) la mụt nguyờn ham cua ham sụ cac trng hp sau: F (x) Phng phỏp: f (x), l mụt nguyờn hm ca hm s ta cn F Â(x) = f (x) chng minh: F (x) = 5x3 + 4x2 - 7x + 120 a) f (x) = 15x2 + 8x - va f (x) = F (x) = ln(x + x + 3) b) x +3 va F (x) = (4x - 5) ìex c) f (x) = (4x - 1) ìex va | THBTN TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN F (x) = tan4 x + 3x - d) f (x) = 4tan5 x + 4tan3 x + va e) ổ x2 + 4ử - 2x ữ ữ F (x) = ln ỗ ỗ f (x) = ữ ỗ ữ ỗ ốx + 3ứ (x + 4) ì(x2 + 3) va F (x) = ln f) x2 - x + x2 + x + va 2(x2 - 1) f (x) = x4 + BT Tim nguyờn ham cua cac ham sụ tha iờu kiờn cho trc cac trng hp sau: f (x), Phng phỏp: Tỡm nguyờn hm ca hm s ũ f (x) ìdx = F (x) +C a) tc i tớnh F (xo ) + C = ììì C Ri sau ú th tỡm hng s f (x) = x3 - 4x + 5, F (1) = S: x4 F (x) = - x2 + 5x - ì 4 b) f (x) = - 5cosx, F (p) = F (x) = 3x - 5sin x + - 3p S: f (x) = c) - 5x2 , F (e) = x F (x) = 3ln x S: 5x2 5e2 + - 2 http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 7| TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN f (x) = d) x2 + , F (1) = ì x F (x) = S: x2 + ln x + f (x) = x x + e) x , F (1) = - F (x) = S: 22 x +2 x ì 5 f) I = ũ sin2x.cosxdx , ổ pử ữ F Âỗ = ỗ ữ ữ ữ ỗ ố3ứ biờt F (x) = - S: 1 cosx - cosx + ì 12 I =ũ g) 3x4 - 2x3 + ìdx, x2 F (x) = x3 - x2 - F (1) = biờt S: + x I =ũ x3 + 3x2 + 3x - ìdx, (x + 1)2 h) F (x) = F (0) = biờt S: x2 +x + ì x +1 x I = ũ sin ìdx, ổ p pữ ữ Fỗ ỗ ữ= ì ỗ ố2ữ ứ i) biờt F (x) = S: x sin x + - ì 2 | THBTN TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN ổ 1ữ ữ I = ũ x ìỗ x + ìdx, ỗ ỗ ữ xữ ố ứ j) F (1) = biờt ì F (x) S: x2 + 3x + + 3ln x + x I =ũ k) 2cos2 x - ìdx, cos2 x ổ p pữ ữ Fỗ ỗ ữ= ì ỗ ố4ữ ứ F (x) = 2x - tan x + biờt S: BT Tim iờu kiờn cua tham sụ m hoc a, b, c F (x) la mụt nguyờn ham cua f (x) : ham sụ F (x) Phng phỏp: l mụt nguyờn hm ca hm s f (x) F Â(x) = f (x) T ú, ta s dng ng nht thc tỡm tham s cn tỡm iù F (x) = mx3 + (3m + 2)x2 - 4x + ù ì i ùù f (x) = 3x2 + 10x - ùợ a) S: m = b) iù F (x) = ln x2 - mx + ùù ùi ì ùù f (x) = 2x + ùùợ x2 + 3x + S: m = - http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 9| TI LIU HC TP c) TON 12 Chuyờn : NGUYấN HM TCH PHN iù F (x) = (ax2 + bx + c) ìex ù ì i ùù f (x) = (x - 3) ìex ùợ a = 0, b = 1, c = - S: d) iù F (x) = (ax2 + bx + c) ìe- 2x ù ì i ùù f (x) = - (2x2 - 8x + 7) ìe- 2x ùợ a = 1, b = - 3, c = S: e) iù F (x) = (ax2 + bx + c) ìe- x ù ì i ùù f (x) = (x2 - 3x + 2) ìe- x ùợ a = - 1, b = 1, c = - S: f) iù ùù F (x) = (a + 1)sin x + b sin2x + c sin3x ì i ùù f (x) = cosx ùùợ S: a = b = c = g) iù F (x) = (ax2 + bx + c) ì 2x - ùù ùi ì ùù f (x) = 20x - 30x + ùù 2x - ợ a = 4, b = - 2, c = S: 10 | THBTN TON HC BC TRUNG NAM TI LIU HC TP h) TON 12 I = ũ (x + 1) ìsin2x ìdx ì Chuyờn : NGUYấN HM TCH PHN I =- S: x +1 cos2x + sin2x + C i) I = ũ x ìe- x ìdx ì I = - (1 + x) ìe- x + C S: I = ũe ìsin x ìdx ì I = x j) S: ex ì(sin x - cosx) +C k) I = ũ x ìcosx ìdx ì S: I = x sin x + cosx + C l) x I = ũ x ìsin ìdx ì S: x x I = - 2x cos + 4sin + C 2 m) I = ũ x ìex ìdx ì S: I = xex - ex + C 78 | THBTN TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN n) I = ũ x ìln(1- x) ìdx ì I = S: x2 ln(1- x) (1 + x)2 ln(1- x) +C 2 I = I = ũ x ìsin x ìdx ì o) S: x2 x sin2x cos2x +C 4 p) I = ũ ln(x + + x2 ) ìdx ì I = x ln(x + + x2 ) - + x2 +C S: I = ũ x ìln q) 1+ x ìdx ì 1- x I =x+ S: x2 - 1 + x ln +C 1- x I =ũ r) lnx ìdx ì x3 I =S: ln x +C 2x 4x2 http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 79 | TI LIU HC TP s) TON 12 I = ũ x ìsin x ìcosx ìdx ì Chuyờn : NGUYấN HM TCH PHN I =- S: 1 x cos2x + sin2x + C t) I = ũe- 2x ìcos3x ìdx ì I = S: - 2x e (3sin3x - 2cos3x) + C 13 I =ũ u) x ìdx ì + cos2x S: 1 I = x tan x + ln cosx + C 2 v) I = ũ x ì(2cos2 x - 1) ìdx ì I = S: x ìsin2x + cos2x + C I = ũ x ìln x ìdx ì w) S: x4 ln x x4 I = +C 16 I =ũ x) 80 | THBTN x ìdx ì sin2 x I = - x cot x + ln sin x + C S: TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN y) I = ũ(x - 2) ìe2x ìdx ì S: 1 I = (x - 2)e2x - e2x + C z) I = ũ x ìln(x2 + 1) ìdx ì I = (x2 + 1) ln(x2 + 1) - x2 - + C S: BT 18 a) Tinh cac nguyờn ham sau: x2 - I = ũ ìln x ìdx ì x ổ 1ử ữ I =ỗ x+ ữ ìln x - x + + C ỗ ữ ỗ ữ xứ x ố S: b) I = ũ cos x ìdx ì S: I = x sin x - 2cos x + C c) I = ũ sin x ìdx ì S: I = - x cos x + 2sin x + C http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 81 | TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN d) I = ũ(8x3 - 2x) ìex ìdx ì 2 I = (4x2 - 1) ìex - 4ex + C S: e) I = ũ x3.ex ìdx ì S: 1 I = x2ex - ex + C 2 I = f) I = ũ x5 ìex ìdx ì S: x3 x3 x e - e +C 3 g) I = ũesinx ìsin2x ìdx ì S: I = 2sin xe sinx - 2esin x + C h) I = ũ x ìe x ìdx ì S: I = 2xe x - xe x + 4e x + C i) I = ũ x ìln(x2 + 1) ìdx ì S: I = (x2 + 1)ln(x2 + 1) - x2 - x + C 82 | THBTN TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN I =ũ j) + ln(x + 1) ìdx ì x2 I =- 1 x - ln x + + ln +C x x x +1 S: k) I = ũex ìln(ex + 1) ìdx ì I = (ex + 1)ln(ex + 1) - ex + C S: I =ũ ln(4x2 + 8x + 3) ìdx ì (x + 1)3 l) 4x2 + 8x + ln 4x2 + 8x + - 4ln x + + C 2(x + 1) S: ổ ữ ữ I = ũỗ + ìln(x + x - 1) ìdx ì ỗ ữ I = (x + x - 1)ln x + x - - x ỗ ữ ỗ ố xứ m) S: x +C C - BI TP TRC NGHIM f (x) = xex Cõu 194 A Mụt nguyờn ham cua ham sụ ex + C la: e ( x - 1) + C e ( x + 1) + C x B x C D x2 x e +C f (x) = (x2 + 2x).ex Cõu 195 Mụt nguyờn ham cua ham sụ la: http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 83 | TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN (2x + 2).ex A B (x2 + x).ex x2ex C D f (x) = xe -x Cõu 196 Cho ham sụ F (x) Mụt nguyờn ham - x -x - (x + 1)e + A (x2 - 2x).ex - (x + 1)e f (x) cua F (0) = tha - x +2 (x + 1)e + B C la: - x (x + 1)e + D f (x) = x sin x f (x) Cho Nguyờn ham cua la: - x cosx + C x sin x + cosx + C A B sin x + x cosx + C - x cosx + sin x + C C D Cõu 197 f (x) = xex Cõu 198 Nguyờn ham cua ham sụ F (x) = 2ex A B la ham sụ: F (x) = ex 2 F (x) = 2x2ex C F (x) = ex + xex D x f (x) = ũ ln tdt f '(x) Cõu 199 A Cho o ham x B la ham sụ nao di õy? lnx C ln x D ln x f (x) = x3ex Cõu 200 Mụt nguyờn ham cua 3 la: x (x3 - 6x + 6)ex (x - 3x + 6x - 6)e A B x (x + 3x - 6)e C D 3x2ex f ( x) = xe x Cõu 201 Mụt nguyờn ham cua ham sụ F ( x) = 2ex F ( x) = 2x2ex A la : B C F ( x) = ex F ( x) = xex + ex 2 D f (x) = (x + 1)sin x Cõu 202 Ham sụ cú cac nguyờn ham la: F (x) = (x + 1)cosx + sinx + C F (x) = - (x + 1)cosx + sinx + C A B F (x) = - (x + 1) cosx - sinx + C F (x) = (x + 1)cosx - sinx + C C D 84 | THBTN TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN f (x) = ln x Cõu 203 Ham sụ cú cac nguyờn ham la: F (x) = F (x) = x(ln x - 1) + C A B +C x F (x) = C ln x +C F (x) = x(ln x + 1) + C D ổ1 ữ f (x) = cosx ỗ + xữ ỗ ữ ữ ỗ ốcosx ứ Cõu 204 Ham sụ cú cac nguyờn ham la: ổx ữ F (x) = sin x ỗ + x2ữ ỗ ữ+C ỗ ốsin x ữ ứ A F (x) = x(1 + sin x) - cosx + C B F (x) = x(1 + sin x) + cosx +C C F (x) = x(1- sin x) + cosx + C D F (x) Cõu 205 Goi ham sụ f (x) = x cos3x la mụt nguyờn ham cua F (0) = , biờt Vy F (x) la: F (x) = A C 1 x sin3x + cos3x + C F (x) = x2 sin3x ũ xe dx B 1 F (x) = x sin3x + cos3x + F (x) = D 1 x sin3x + cos3x + 9 x Cõu 206 Tinh , ta c kờt qua la: F ( x) = ex xex + C A F ( x) = ex + xex + C B F ( x) = - e xe + C x C Cõu 207 F ( x) = - ex + xex +C x D Tinh ũ x cosxdx , ta c kờt qua la: F ( x) = x sin x + cosx + C A F ( x) = x sin x - cosx + C B F ( x) = - x sin x + cosx + C C Cõu 208 F ( x) = - x sin x - cosx +C D Tim ũ x cos2xdx la: http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 85 | TI LIU HC TP A C TON 12 Chuyờn : NGUYấN HM TCH PHN 1 x sin2x + cos2x +C B x2 sin2x +C D 1 x sin2x + cos2x + C 2 sin2x +C f (x) = ( x2 + 2x) ex Cõu 209 Mụt nguyờn ham cua ham sụ la x x F (x) = (2x + 2).e F (x) = x e A B F (x) = (x2 + x).ex F (x) = (x2 - 2x).ex C D f (x) = x2ex Cõu 210 Mụt nguyờn ham cua la F (x) = F (x) = (x2 - 2x + 2)ex A B x F (x) = 2xex F (x) = (x - 2x - 2)e C D F ( x) Cõu 211 Nguyờn ham F ( 0) = f (x) = xe- x cua tha F (x) = - ( x + 1) e + A B F (x) = ( x + 1) e- x + C Kờt qua nao sai cac kờt qua sau ? A Cõu 213 ũ xe dx = x 86 | THBTN B ũ x sin xdx = - x cosx + sin x +C ũ x sin2xdx = D - x cos2x + sin2x + C Kờt qua nao sai cac kờt qua sau ? A Cõu 214 - x2.cosx +C ũ x cosxdx = x sin x + cosx +C 3x ũ xe dx = C F (x) = ( x + 1) e- x + D ũ x sin xdx = C la F ( x) = - ( x + 1) e- x + -x Cõu 212 x3 x e xe3x 3x - e +C x B x x x e + C ũ xe dx = xe ũe x D dx = x - ex +C -x - x +C ex e Kờt qua nao sai cac kờt qua sau ? TON HC BC TRUNG NAM TI LIU HC TP TON 12 A Chuyờn : NGUYấN HM TCH PHN C Cõu 215 ũx dx = B - ln x +C x x ũe dx = ln x D -x +C 2x 2e 4e2x ũx dx = ln3 x +C - ln x +C 2x2 4x2 ũ xe - x B 3x xe 3x ũ xe dx = - 9e +C 3x Cõu 217 ũ ln xdx = Kờt qua nao sai cac kờt qua sau ? 2x C D ũ ln xdx = x ln x - 2( x ln x - x) +C x A x3 x3 ũ x ln xdx = ln x - +C 2 Cõu 216 ũ lnxdx = x +C Kờt qua nao sai cac kờt qua sau ? ln x C B x x ũ x lnxdx = ln x - +C A ũ ln xdx = x ln x - x +C dx = - xe- x - e- x +C x2 2x ũ xe dx = e +C 2x D Kờt qua nao sai cac kờt qua sau ? ũ x lnxdx = A x3 +C x x3 x3 ũ x ln xdx = ln x - + C B ũ ln( x + ) ( ) 1+ x2 dx = x ln x + + x2 - + x2 + C C ũe x sin xdx = D ex ( sin x - cosx) +C f (x) = x.sin ( 2x + 1) Cõu 218 Tim nguyờn ham cua ham sụ ũ f (x)dx = A B x2 ũ f (x)dx = - cos( 2x + 1) +C x C x cos( 2x + 1) + sin ( 2x + 1) + C ũ f (x)dx = cos( 2x + 1) - sin ( 2x + 1) + C http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 87 | TI LIU HC TP TON 12 ũ f (x)dx = D Chuyờn : NGUYấN HM TCH PHN x cos( 2x + 1) + sin ( 2x + 1) + C 2 f (x) = x.ln( + x) Cõu 219 Tim nguyờn ham cua ham sụ x2 ũ f (x)dx = 2(x + 1) +C A B x2 ũ f (x)dx = ln( 1+ x) - x ln(1+ x) +C C ũ f (x)dx = 2( x ũ f (x)dx = D ) - ln ( + x) - x x + +C x2 x ln( + x) - x2 + ln(x + 1) + C 2 I = ũ cos2x.ln(sin x + cosx)dx Cõu 220 Nguyờn ham cua ham sụ: A F(x) = B F(x) = C F(x) = D F(x) = Cõu 221 la: 1 + sin2x) ln( + sin2x) - sin2x + C ( 1 + sin2x) ln( + sin2x) - sin2x + C ( 1 + sin2x) ln ( + sin2x) - sin2x + C ( 4 1 + sin2x) ln ( 1+ sin2x) + sin2x + C ( 4 Nguyờn ham cua ham sụ: - A F(x) = - I = ũ( x - 2) sin3xdx ( x - 2) cos3x + sin3x + C la: ( x - 2) cos3x + sin3x + C B F(x) = ( x + 2) cos3x + sin3x +C - ( x - 2) cos3x + sin3x +C 3 C F(x) = D F(x) = BI TP TRC NGHIM DNG TON KHC ( C THấM ) Cõu 222 88 | THBTN Trong cac mờnh sau, mờnh nao sai? TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN x2 + 6x + F (x) = 2x - A Ham sụ ham sụ va F (x) = + 2sin2 x B Ham sụ ham sụ x2 + 10 G (x) = 2x - l nguyờn ham cua cựng mụt G (x) = 1- cos2x va la nguyờn ham cua cựng mụt F (x) = x2 - 2x + C Ham sụ D Ham sụ A B C f (x) = cos x Cac mờnh sau, mờnh nao SAI? ũkf ( x) dx = kũ f ( x) dx ( k ẻ R ) ũ f ( x) g( x) dx = ũ f ( x) dx ũ g( x) dx ũ ộờởf ( x) + g( x) ựỳỷdx = ũ f ( x) dx + ũ g( x) dx f m+1 ( x) ũ f ( x) f Â( x) dx = m +1 D +C F ( x) = ( a sin x + bcosx) ex tri cua f ( x) = cosxe x la mụt nguyờn ham cua A a = 0,b = Cho ham sụ C xac inh trờn f ( x) cua ham sụ trờn K K Ham sụ c goi la nguyờn ham nờu: F ( x) f ( x) , " x ẻ K B f ( x) = F ( x) , " x ẻ K C A D F ( x) F ( x) = f ( x) , " x ẻ K A a =b = a =b = B f ( x) Cõu 226 thi gia a b , la : a = 1,b = Cõu 225 la nguyờn ham cua ham sụ m Cõu 224 x2 - 2x + la nguyờn ham cua ham sụ F (x) = sin x Cõu 223 x- f (x) = f ( x) F ( x) , " x ẻ K D Cac tinh chõt nguyờn ham sau õy tinh chõt nao sai? ũ f '(x)dx = f (x) +C B ũ K f (x)dx = K ũ f (x)dx ( K http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 0) 89 | TI LIU HC TP C Cõu 227 TON 12 ũ ộờởf (x) g(x)ựỳỷdx = ũ f (x)dx ũ g(x)dx ũ F (x)dx = f (x) + C D Trong cac mờnh sau, tim mờnh sai ũ f '(x)f (x)dx = A C Chuyờn : NGUYấN HM TCH PHN f 3(x) +C ũ ộờởf (x).g(x)ựỳỷdx =ũ f (x)dx.ũ g(x)dx B ũ ộởờf (x) + g(x)ựỷỳdx =ũ f (x)dx +ũ g(x)dx ũ kf (x)dx =kũ f (x)dx D ( k la hng sụ) f (x) = (ax2 + bx + c) 2x - Cõu 228 Nờu g(x) = la 2x - trờn khoang B A Cõu 229 ổ ỗ ỗ ; +Ơ ỗ ố2 10x2 - 7x + mụt ữ ữ ữ ữ ứ thi C nguyờn a +b + c ham cua ham sụ cú gia tri la D g(x) = (ax2 + bx + c) 2x - a b c Xac inh , , cho la mụt nguyờn ham f (x) = ổ ỗ ỗ ; +Ơ ỗ ố2 20x2 - 30x + cua ham sụ a = 4, b = 2, c = A a = - 2, b = 1, c = C 2x - ữ ữ ữ ữ ứ khoang a = 1, b = - 2, c = B a = 4, b = - 2, c = D F (x) = ex 2(a tan2 x + b tan x + c) Cõu 230 Tim nguyờn ham f (x) = ex la mụt nguyờn ham cua ổ p pữ ỗ ữ ; ỗ ữ ỗ ố 2ữ ứ tan3 x trờn khoan x F (x) = e ổ ỗ ỗ tan2 x ỗ ỗ ỗ ố2 2ữ ữ tan x + ữ 2ữ ữ ứ A x F (x) = e ổ ỗ ỗ tan2 x ỗ ỗ ỗ ố2 1ử ữ tan x + ữ ữ ữ 2ứ ữ B ổ 1ử ữ ỗ1 ữ F (x) = ex ỗ tan x + tan x + ữ ỗ ữ ỗ 2 ữ ỗ ố ứ C ổ ỗ1 F (x) = ex ỗ tan2 x ỗ ỗ ỗ ố2 tan x 2ử ữ ữ ữ ữ 2ứ ữ D F (x) = ( ax2 + bx + c) e- x Cõu 231 Nờu la f ( x) = ( - 2x2 + 7x - 4) e- x nguyờn ham cua ham sụ ( a;b;c) thi 90 | THBTN mụt bng bao nhiờu? TON HC BC TRUNG NAM TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN ( 2;- 3;1) ( 1;3;2) A B Cõu 232 A ( 1;- 1;1) Cho C ũ f ( x) dx = F ( x) +C Khi ú, vi F ( ax + b) + C 2a D aạ , ta cú ũ f ( ax + b) dx bng F ( ax + b) + C a B F ( ax + b) +C a.F ( ax + b) + C C D Cõu 233 A ( - 2;7;- 4) Cho iù cosxe sin x ; " x < ùù f ( x) = ùi ùù ;" x ùùợ + x iù esinx ; " x < ù F ( x) = ùi ùù + x - 1; " x ùợ iù e ; " x < ù F ( x) = ùi ùù + x - 1; " x ùợ Nhn xột nao sau õy ỳng ? f ( x) la mụt nguyờn ham cua cosx B C D iù esin x ; " x < ù F ( x) = ùi ùù 1+ x; " x ùợ f ( x) la mụt nguyờn ham cua f ( x) la mụt nguyờn ham cua iù ecosx ; " x < ù F ( x) = ùi ùù + x; " x ùợ f ( x) la mụt nguyờn ham cua ũ( x - 2) sin3xdx = Kờt Mụt nguyờn ham S = ab +c tng bng S = 15 S = 14 A B ( x - a) cos3x b Cõu 234 f ( x) = Cõu 235 x> C S =3 20x2 - 30x + 2x - D ( thi S = 10 ) 2x - vi F ( x) ham sụ sin3x + 2017 c ;F ( x) = ax2 + bx + c Cho cac ham sụ + f ( x) la mụt nguyờn am cua ham sụ thi cac gia tri cua a,b,c la http://dethithpt.com Website chuyờn thi, file word cú li gii 0982.56.33.65 91 | TI LIU HC TP TON 12 Chuyờn : NGUYấN HM TCH PHN a = 4;b = 2;c = - A a = 4;b = 2;c = B a = 4;b = - 2;c = - C 92 | THBTN a = 4;b = - 2;c = D TON HC BC TRUNG NAM ... tan x - x + A F (x) = x - tan x + C Cõu 74 Tinh - A Cõu 75 A C Cõu 76 ũ(cos6x - F (x) = tan x - x - p F (x) = x - tan x - p B p D 1)dx , kt qu la: cos(3x - 1) + C Tim - ũ sin(3x - p B cos(3x - 1)... 20x3 + 8x - x5 + x3 + C f (x) = ( x - 3) ( x - 3) - 20x3 + 8x +C 4( x - 3) ( x - 3) B ( x - 3) C D 3 f (x) = x3 - 3x2 + Cõu Mụt nguyờn ham cua ham sụ la 2 A 3x - 6x B 3x - 6x + C C x - x3 + 5x... ex - e- x Cõu 98 Tim nguyờn ham cua ham sụ ũ f ( x) dx = e x A ũ f ( x) dx = e x C ũ f ( x) dx = - e x -x + e +C B - x ũ f ( x) dx = - e x - e +C D + e- x + C - x - e +C f (x) = 2x. 3- 2x