Evaluate `int_(0)^(4)(x^2+2x+8)dx`

Answer

D

Explanation

`lim_(xtoa)(int_(a)^(x)f(x)dx-((x-a)/2)(f(x)+f(a)))/((x_a)^(3))=0`
or `lim_(hto0)(int_(a)^(a+h)f(x)dx - h/2 (f(a+h)+f(a)))/(h^(3))=0`
or `lim_(hto 0) (f(a+h)-1/2 [f(a)+f(a+h)]-h/2(f'(a+h)))/(3h^(2))=0`
[Using L' Hospital's rule]
or `lim_(hto0) (1/2f(a+h)-1/2f(a)-h/2f'(a+h))/(3h^(2))=0`
or `lim_(h to0)(1/2)f'(a+h)-1/2f'(a+h)-h/2f''((a+h))/(6h)=0`
[Using L'Hospital's rule]
or `lim_(hto0)(-f''(a))=0AAaepsilonR`
Thus `f(x)` must be of maximu degree 1.