大学高数问题,求4,5题详解,和前三的答案

4,
S(1-x)dx/(4-x^2)^(1/2)
= Sdx/(4-x^2)^(1/2) - Sxdx/(4-x^2)^(1/2)
= (1/2)Sdx/[1-(x/2)^2]^(1/2) - (1/2)Sd(x^2)/(4-x^2)^(1/2)
= Sd(x/2)/[(1-(x/2)^2]^(1/2) + (1/2)Sd(4-x^2)/(4-x^2)^(1/2)
= arcsin(x/2) + (4-x^2)^(1/2) + C
其中,C为任意常数.
5,
特征方程为,0 = r^2 - 4r + 5 = r^2 - 4r + 4 + 1 = (r-2)^2 + 1,
r = 2 + i 或r = 2 - i .
通解为y = e^[2x][c(1)cos(x) + c(2)sin(x)]
其中,c(1),c(2)为任意常数.
1,
0 < x^2 - 3x - 10 = (x-5)(x+2),x>5 或x5.
2,
2/3
3,
dy = 1/sin(3x)*3dx = 3dx/sin(3x)