f(x)=3x^2*∫[0,1]f(t)dt+x∫[0,1]{f'(t)}^2dt+∫[0,1]f(t)dt
f(-x)=3x^2*∫[0,1]f(t)dt-x∫[0,1]{f'(t)}^2dt+∫[0,1]f(t)dt
f(x)+f(-x)=6x^2*∫[0,1]f(t)dt+2∫[0,1]f(t)dt
f(x)-f(-x)=2x∫[0,1]{f'(t)}^2dt
f'(x)-f'(-x)=12x∫[0,1]f(t)dt
f'(x)+f'(-x)=2∫[0,1]{f'(t)}^2dt
f(x)-f(-x)=x(f'(x)+f'(-x))
f(x)-f(-x)=Ax
f(x)=3x^2*∫[0,1]f(t)dt+Ax/2+∫[0,1]f(t)dt
f'(x)=6x∫[0,1]f(t)dt+A/2
f''(x)=6∫[0,1]f(t)dt
C1=∫[0,1]f(t)dtとすると
f''(x)=6C1
f'(t)=6C1x+C2
f(t)=3C1x^2+C2x+C3
f(0)=∫[0,1]f(t)dt=C1+C2/2+C3=C3 ∴C2=-2C1
∫[0,1]f(t)dt=2C1+2C3=C1 ∴C3=-C1/2
f(x)=C(6x^2-4x-1)
f(x)=3x^2*∫[0,1]C(6x^2-4x-1)dt+x∫[0,1]{(C(6x^2-4x-1))'}^2dt+∫[0,1]C(6x^2-4x-1)dt
=6Cx^2+16C^2*x-C
C(6x^2-4x-1)=6Cx^2+16C^2*x-C ∴C=-1/4, 0
f(x)=(-6x^2+4x+1)/4, 0