If f (x) is a function satisfying f ^ (x)= f (x) with f (0)=1 and g (x) is a function that satisfies f (x)+ g (x)=x^2 . Then the value of the integral ₀^1 f (x) g (x) d x is

Step-by-step solution

As f ^ (x)= f (x) f^ (x) f (x) =1 Integrating on both sides, we get f (x)=x+ c As f(0)=1 array ll & ( i ) c =0 & f (x)=x & f (x)= e ^x array aligned & As f (x)+ g (x)=x^2 & ~g (x)=x^2- e ^x aligned f (x) g (x)= e ^x (x^2- e ^x ) aligned & = ₀^1 ( e ^x x^2- e ^ 2 x ) d x & = [ (x^2-2 x+2 ) e ^x ]₀^1- 1 2 e ^2+ 1 2 & = e - 1 2 e ^2- 3 2 aligned