Let f(x) and g(x) be continuous on R (set of real numbers). If _ x 0 f(x) ^2 x =8, _ x 0 g(x) 2 x-x e^x+x^3+x-2 = and _ x 0 (1+2 f(x))^ 1 / g(x) = 1 e , then find the value of .

Step-by-step solution

_ x 0 f(x) ^2 x = _ x 0 f(x) x^2 ^2 x x^2 = _ x 0 f(x) x^2 =8 and _ x 0 f(x)=0 and _ x 0 g(x) 2 (1- x^2 2 ! + )-x (1+x+ x^2 2 ! + )+x^3+x-2 = gathered = _ x 0 g(x) x^2 (-2+ x 2 + ) =1 _ x 0 g(x) -2 x^2 = _ x 0 (1+2 f(x))^ 1 / g(x) =e^ _ x 0 2 f(x) / x^2 g(x) / x^2 =e^ -8 / = 1 e =8 gathered