答案:A

6.已知一物体自由下落的速度为v=gt,则当t从2 s至3 s时,物体下落的距离为　　　　　.

解析:物体下落的距离s=∫_2^3▒ g(t)dt=(1/2 gt^2 ) "|" _2^3=1/2g(32-22)=5/2g.

答案:5/2g

7.已知函数f(x)=3x2+2x+1,若∫_("-" 1)^1▒ f(x)dx=2f(a)成立,则a=　　　　　.

解析:∵∫_("-" 1)^1▒ f(x)dx=∫_("-" 1)^1▒ (3x2+2x+1)dx=(x3+x2+x)"|" _("-" 1)^1=4,

　　∴2(3a2+2a+1)=4,即3a2+2a-1=0,解得a=-1或a=1/3.

答案:-1或1/3

8.已知2≤∫_1^2▒ (kx+1)dx≤4,则实数k的取值范围是　　　　　.

解析:∵∫_1^2▒ (kx+1)dx=(1/2 kx^2+x) "|" _1^2=3/2k+1,

　　∴2≤3/2k+1≤4.∴2/3≤k≤2.

答案:2/3≤k≤2

9.计算下列定积分.

(1)∫^e▒ _0 3/(3x+2)dx;(2)∫_("-" π/2)^(π/2)▒ (2x+cos x)dx;

(3)∫_0^2▒ (x2-1)dx.

解(1)∵[ln(3x+2)]'=3/(3x+2),

　　∴∫^e▒ _0 3/(3x+2)dx=ln(3x+2)〖"|" ^e〗_0

　　=ln(3e+2)-ln(3×0+2)=ln(3e+2)/2.

　　(2)∵(sin x+x2)'=cos x+2x,

　　∴∫_("-" π/2)^(π/2)▒ (2x+cos x)dx=(sin x+x2)"|" _("-" π/2)^(π/2)

　　=sinπ/2+(π/2)^2-[sin("-" π/2)+("-" π/2)^2 ]=2.

　　(3)∵y=|x2-1|={■(x^2 "-" 1"," 1≤x≤2"," @1"-" x^2 "," 0≤x<1"," )┤

　　∴∫_0^2▒ (x2-1)dx=∫_0^1▒ (1-x2)dx+∫_1^2▒ (x2-1)dx

　　=(x"-" 〖x/3〗^3 ) "|" _0^1+(x^3/3 "-" x) "|" _1^2

　　=(1"-" 1/3)+(8/3 "-" 2)-(1/3 "-" 1)=2.

10.已知f(x)在R上可导,f(x)=x2+2f'(2)x+3,试求 ∫_0^3▒ f(x)dx的值.

解∵f(x)=x2+2f'(2)x+3,∴f'(x)=2x+2f'(2).

　　∴f'(2)=4+2f'(2).∴f'(2)=-4.

　　∴f(x)=x2-8x+3.

　　∴∫_0^3▒ f(x)dx=(1/3 x^3 "-" 4x^2+3x) "|" _0^3=-18.

B组