The value of _ x 0 e^2 - (1+x+ x^2 2 )^ 2 x x^2 is equal to

Step-by-step solution

Let L = _ x 0 e^2 - (1+x+ x^2 2 )^ 2 x x^2 . We can rewrite the term (1+x+ x^2 2 )^ 2 x as e^ 2 x (1+x+ x^2 2 ) . Using the Maclaurin series expansion for (1+u) where u = x+ x^2 2 : (1+u) = u - u^2 2 + u^3 3 - Substituting u = x+ x^2 2 : (1+x+ x^2 2 ) = (x+ x^2 2 ) - 1 2 (x+ x^2 2 )^2 + 1 3 (x+ x^2 2 )^3 - Expanding up to x^3 : (1+x+ x^2 2 ) = x + x^2 2 - 1 2 (x^2 + x^3 ) + 1 3 (x^3 ) + O(x^4) = x + x^2 2 - x^2 2 - x^3 2 + x^3 3 + O(x^4) = x - x^3 6 + O(x^4) Now, multiply by 2 x : 2 x (1+x+ x^2 2 ) = 2 x (x - x^3 6