Let f, g : (0, ) R be two functions defined by f(x) = _ - x ^ x |t| (1+t^2) dt and g(x) = ₀^x t 1+t dt . Then the value of e^ 1 3 (f(3) + g(3)) is equal to

Step-by-step solution

Given f(x) = _ - x ^ x |t| (1+t^2) dt . Since the integrand is an even function, we can write: f(x) = 2 ₀^ x t (1+t^2) dt Let t^2 = u 2t dt = du . The upper limit changes from x to x . f(x) = ₀^x (1+u) du We are also given g(x) = ₀^x t 1+t dt . Adding the two functions: f(x) + g(x) = ₀^x ( (1+t) + t 1+t ) dt Notice that the integrand is the exact derivative of t (1+t) . f(x) + g(x) = [ t (1+t) ]₀^x = x (1+x) Substitute x = 3 : f(3) + g(3) = 3 (1+3) = 3 4 = (4^3) = 64 We need to find the value of e^ 1 3 (f(3) + g(3)