Lim g (x)/x = lim g' (x) = f (0), so it is assumed that x = 0 in the generalized integral is not deficient.
In addition, limg2 (x)/x = lim2gg' (x) = 2g (0) * g' (0) = 0.
Therefore, for any X>0, it is
Integral (from 0 to x) g (x)/x 2dx = integral (from 0 to x) g 2 (x) d (- 1/x)
=-g 2 (x)/x | upper limit x lower limit 0+ integral (from 0 to X)2g(x)g'(x)/xdx
Because -g 2 (x)/x
& lt=2 integer (from 0 to X)g(x)/x *f(x) dx.
According to Cauchy-Schwartz inequality
& lt=2 integral (from 0 to x) g 2 (x)/x 2dx (1/2) * integral (from 0 to x) f 2 (x) dx (1/2)
In order to solve this inequality
Integral (from 0 to x) g 2 (x)/x 2dx < =4 Integral (from 0 to x) f 2 (x) dx,
So the generalized integral converges and the inequality is established.
Final summary of excellent school teachers 1
This school year, I always adhere to the party's education and teaching policy, face all students, teach
1. What are the subjects of the correspondence undergraduate entrance examination?
Correspondence candidates need to take four cultural courses: Chinese, Mathemat