Let f be a continuous function defined on [0, ) such that g(x) = ₀^x t f(t) dt . If g(x^2) = x^4 ( x^2 2 ) , then the value of ₀^2 f(x) dx is :

Step-by-step solution

Given g(x^2) = x^4 ( x^2 2 ) Let x^2 = u , then g(u) = u^2 ( u 2 ) . We are given g(x) = ₀^x t f(t) dt . By Newton-Leibniz formula, g'(x) = x f(x) . Differentiating g(u) with respect to u : g'(u) = 2u ( u 2 ) + u^2 ( 2 ) ( u 2 ) Equating this to u f(u) : u f(u) = 2u ( u 2 ) + 2 u^2 ( u 2 ) For u > 0 , f(u) = 2 ( u 2 ) + 2 u ( u 2 ) Now, evaluate the integral ₀^2 f(x) dx : ₀^2 f(x) dx = ₀^2 2 ( x 2 ) dx + ₀^2 2 x ( x 2 ) dx Applying integration by parts on the second term: ₀^2 2 x ( x 2 ) dx = [ x ( x 2 ) ]₀^2 - ₀^2