2012考研 数学 高数题 求解

3.(2) 猜测 x1 < x2 x = 0.69482,x2 = 0.76817
sin(cos(x1)) = x1 cos(sin(x2)) = x2
=> x1 - x2 = sin(cos(x1)) - cos(sin(x2)) = sin(cos(x1)) - sin(π/2 - sin(x2))
= 2 cos [ cos(x1) + π/2 - sin(x2)] sin[ cos(x1) - π/2 +sin(x2) ]
4.I = ∫ [0,1] .+ ∫ [1,+∞) .
换元,令 t = 1/x,∫ [1,+∞) .= ∫ [0,1] t ^(2-k) / (1+t²)² dt
I = ∫ [0,1] [ x^k + x^(2-k) ] / (1+x²)² dx
比较,k=2,I = .= π/4,k = 1,I = .= 1/2
k = 3/2 和 1/2 时,I 相等.
故 选择 C.