The value of the integral ₀² x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) , dx is equal to:

Step-by-step solution

Let I = ₀² x(x^2 + x + 1) x + 1 , x^4 + x^2 + 1 , dx . Using the identity x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1) : I = ₀² x , x^2 + x + 1 x + 1 , (x^2 + x + 1)(x^2 - x + 1) , dx Cancelling x^2 + x + 1 : I = ₀² x (x + 1)(x^2 - x + 1) , dx Using (x + 1)(x^2 - x + 1) = x^3 + 1 : I = ₀² x x^3 + 1 , dx Substitution: Let t = x^ 3/2 , so dt = 3 2 x^ 1/2 , dx x , dx = 2 3 , dt . Also, t^2 = x^3 , so x^3 + 1 = t^2 + 1 . Changing limits: x = 0 t = 0 x = 2 t = 2^ 3/2 = 2 2 Therefore: I = 2 3 ₀^ 2 2 dt t^2 + 1 Using dt t^2