F(x)为f(x)的原函数,当x>0时,f(x)F(x)=sin²2x,且F(0)=1,F...

问题描述:

F(x)为f(x)的原函数,当x>0时,f(x)F(x)=sin²2x,且F(0)=1,F...
F(x)为f(x)的原函数,当x>0时,f(x)F(x)=sin²2x,且F(0)=1,F(x)≧0,求f(x)

f(x)F(x)=sin²2x,因为F'(x)=f(x),sin²2x=(1-cos4x)/2
对方称两边积分得到:[F(x)]^2=x-sin4x/4+c,又因为F(0)=1,得到c=1
故F(x)=√(4x+4-sin4x)/2