(设f(x)为连续函数,试证：∫0到a,x^3f(x^2)dx=1/2∫0到a^2xf(x)dx,a>0∫0到x,[∫0到tf(x)dx]dx=∫0到xf（t)(x-t)dt

Question

(设f(x)为连续函数,试证：∫0到a,x^3f(x^2)dx=1/2∫0到a^2 xf(x)dx,a>0
∫0到x,[∫0到tf(x)dx]dx=∫0到x f（t)(x-t)dt

Answer 1

1 ∫[0,a] x^3f(x^2)dx 令x^2=t
= ∫[0,a^2] t^(3/2) f(t) d √t
=1/2 ∫[0,a^2] t f(t) d t 因d √t =1/(2√t) t换为x
=1/2 ∫[0,a^2] x f(x) d x
命题得证