Let a function g(x) be defined as g(x) = _ e^x ^ e^ 10-x 1 y ( ( _e y)^3 ( _e y)^3 + (10 - _e y)^3 ) dy for 0 x 5 . The value of ₀^5 6 (g(x))^2 dx is

Step-by-step solution

For the function g(x) , substitute t = _e y dt = 1 y dy . When y = e^x , t = x . When y = e^ 10-x , t = 10-x . g(x) = _x^ 10-x t^3 t^3 + (10 - t)^3 dt ... (i) Using the property _a^b f(t) dt = _a^b f(a+b-t) dt , the sum of the limits is x + (10 - x) = 10 . g(x) = _x^ 10-x (10-t)^3 (10-t)^3 + (10 - (10-t))^3 dt g(x) = _x^ 10-x (10-t)^3 (10-t)^3 + t^3 dt ... (ii) Adding (i) and (ii): 2 g(x) = _x^ 10-x t^3 + (10-t)^3 t^3 + (10-t)^3 dt = _x^ 10-x 1 dt 2 g(x) = (10 - x) - x = 10 - 2x g(x) = 5 - x Now, evaluate the requi