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

设中间变量t=x^2（t>0），当x=0时,t=0,当x=a时,t=a^2.
x^3=t^(3/2),x^2=t,dx=d(t^(1/2))=t^(-1/2)dt.
将等式左边带入上述数据，再将变量t替换为x即可得右边。

Answer 2

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

Answer 3

∫0到a，x^3f(x^2)dx=∫0到a，x2f(x^2)*xdx=1/2∫0到a，x²3f(x^2)dx²，令t=x²，则
1/2∫0到a，3f(x^2)dx²=1/2∫0到a² tf(t)dt=1/2∫0到a^2 xf(x)dx