Giáo án Giải tích 12

CHƯƠNG3 : NGUYÊN HÀM - TÍCH PHÂN VÀ ỨNG DỤNG BÀI 1 : NGUYÊN HÀM I .Định nghĩa và ký hiệu: Định ngĩa : F(x) là một nguyên hàm của hàm số f(x) nếu F’(x) = f(x) Ký hiệu:fx.dx=f(x) Định lí : fx.dx=Fx+C II. Tính chất: K.fx.dx=Kfx.dx fx±gx.dx=fxdx±gxdx tChú ý : Nguyên hàm dạng tích , và hửu tỷ không có công thức phải biến đổi đưa về tổng hiệu III .Công thức: Nhóm 1: tK.dx=Kx+C t0.dx=C txα.dx=xα+1α+1+C t(kx)α.dx=1k (kx)α+1α+1 + C (α≠-1) tdxx=lnx+ C tdxkx=1k lnkx+C 2 . Nhóm II: tsinx.dx= -cosx+C tsinkx.dx=-1kcoskx+C tcosx.dx=sinx+C tcoskx.dx=1ksinkx+C td xcos2x=tanx+C tdxcos2kx=1ktankx+C tdxsin2x= -cotx+C tdxsin2(kx)= -1kcotkx+C Nhóm III: taxdx=axlna+C takxdx=1kakxlna+C texdx=ex+C tekxdx=1kekx+C \ BÀI TÂP TRONG SÁCH GIÁO KHOA [1/100-CB] Trong cập các hàm số dưới đây, hàm số nào là một nguyên hàm của hàm số còn lạị? a.ex và -e-x b.sin2x và sin2x c.1-2x2exvà 1-4x2ex Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ [2/100-CB]Tìm nguyên hàm của các hàm số sau: a.f(x) = x+ x+13x b. f(x)=2x-1ex c. f(x)=1sin2x.cos 2x d.f(x)=sin5x.cos3x e.f(x)=tan2x g.e3-2x h.f(x)=11+x(1-2x) Giải 3. [1/141-NC] Tìm nguyên hàm của các hàm số sau: a. f(x)=3x2 + x2 b. f(x)=2x3 + 5x + 7 c. f(x)= 1x2 – x2 - 13 d. f(x)= x13 e. f(x)=102x Giải 4. [2/141-NC] Tìm: a.(x + 3x )dx b.xx+xx2dx c.4sin2xdx d.1+cos4x 2dx Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ 5. [3/141-NC] Chọn khẳng định đúng ? Nguyên hàm của hs y=xsinx là (A) x2 sinx2 +C (B) –xcosx +C (C) –xcosx + sinx +C Giải 6. [4/141-NC] Khẳng định sau đúng hay sai ? Nếu f(x)=(1 – x )` thì fxdx= -x +C Giải -------------------------------------------------------------------------------------------------------------------- BÀI TẬP LÀM THÊM Tìm họ nguyên hàm của các hàm số sau ( nhóm 1) : 1 (x2-2x+3)dx 2.1-x2 - 3x-3) dx 3(12x+12x2-5x5)dx 4 (x2+12)dx 5.x+3x2-5x3dx 6.(1x-13x)dx 7 xdx 8 4x2-4x+15dx 9dxx+9-x 10.x+!(x-1)3xdx 11.xxdx 12.2x+3x-2dx 13.dxx(x+1) 14.dxx-1(x-2) 15.x4+x-4+2x3dx 16.x4+9x-4+6xxdx 17x-2x+1dx 18. x-1x+1dx 19dxx2-2x+1 20.x2-x+3x-2dx 21. x2-x+4x-2dx 22. dxx2-4x+3 23x+1x3-3x+2dx 24.(2x+3)2x+1dx Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ 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........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ 2.Tìm họ nguyên hàm của các hàm số sau (nhóm 2) 1.1+cos3cos2dx 2. sin2xcosxdx 3. tan2xdx 4.cos2xdx 5. sin2xdx 6.cos2xsin2xdx 7.sin3xdx 8.cos3xdx 9. (cosx+sinx)2dx 10. sin4x+cos4xdx 11.sin6x2x+cos62xdx 12.4sin3xcosx.dx Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ 3.Tìm họ nguyên hàm của các hàm số sau (nhóm 3) : 1. 32x+2xdx 2. ex+e-xdx 3. (23x.52x)dx 4. ex+3x.exdx 5.ex2+e-xcos2 xdx 6.e2-5x+1exdx 7. 22x3x7xdx 8. ex(1-e-xx3)dx 9. ex3xdx Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... 4. a. Tìm nguyên hàm f(x)của hàm số f(x)=1 - 2x. Biết răng f(1)= 2 b. Tìm nguyên hàm f(x)của hàm số f(x)=x2 - 4x + 5 . Biêt rằng f(0)= - 3 c. Tìm nguyên hàm f(x) của g(x) = e2x +sin3x . Biêt rằng f(0)= 1 Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ 5. Tìm nguyên hàm của hàm số sau : a. f(x) =(x – 9)4 b. f(x)=1(2-x)2 c.f(x) = x 1-x2 d. f(x)= 1 2x+1 e. f(x) = 1-cos2x cos2x f. f(x) = 2x+1x2+x+1 Giải ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ....................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ .........................................................................................................