If f and g are two functions such that 2 f(1)=g(2)=4 and 2 f(9)=g(10)=20 and ₀^2 (x^2 g (f (x^3+1 ) ) f^ (x^3+1 )-3 x^2 ) d x=0 , then find the value of ₄²⁰ g⁻¹(x) d x .

Step-by-step solution

aligned & f (x^3+1 )=t f^ (x^3+1 ) 3 x^2 d x=d t & 1 3 ₂¹⁰ g(t) d t- ( 3 x^3 3 )₀^2=0 & ₂¹⁰ g(t) d t=24 ₂¹⁰ g(x) d x+ ₄²⁰ g⁻¹(x) d x=200-8 & ₄²⁰ g⁻¹(x) d x=200-8-24=168 aligned