Let f(x) = ₀^ x^2 t 2 t dt and g(x) = ₀^x t ^2 t dt . If x ( 0, 2 ) satisfies the equation f(x) + g(x) = 4 , then the value of 48 x is equal to

Step-by-step solution

Given f(x) = ₀^ x^2 t 2 t dt . Let t = u 1 2 t dt = du . When t = 0 , u = 0 . When t = x^2 , u = x . f(x) = ₀^x u du We are also given g(x) = ₀^x t ^2 t dt . Adding f(x) and g(x) : f(x) + g(x) = ₀^x ( t + t ^2 t) dt Notice that the integrand is the exact derivative of t t with respect to t . f(x) + g(x) = [ t t ]₀^x = x x We are given that f(x) + g(x) = 4 . x x = 4 By inspection, for x ( 0, 2 ) , x = 4 satisfies the equation since 4 ( 4 ) = 4 1 = 4 . We need to find the value of 48 x : 48 4 = 12 Answer: 12