Let f be a real valued continuous function defined on the positive real axis such that g(x)= ₀^x t f( t ) dt . If g (x^3 )=x^6+x^7 , then value of _ r=1 ¹⁵ f ( r ^3 ) is :

Step-by-step solution

g(x)=x^2+x^ 7 / 3 aligned & g^ (x)=2 x+ 7 3 x^ 4 / 3 & f(x)= g^ (x) x & f(x)=2+ 7 3 x^ 1 / 3 & f (r^3 )=2+ 7 3 r & _ r=1 ¹⁵ (2+ 7 3 r )=2(15)+ 7 3 ( 15(16) 2 ) aligned =310