Question

What is the average value of y=cosx/x^2+x+2 on the closed interval "[-1, 3]"?

Answer 1

We have 
y=cos x/(x²+x+2)  on the closed interval [-1, 3]

we know that
The average value of f(x) on the interval [a, b] is given by: 
F(avg) = 1/(b - a) ∫ f(x) dx (from x=a to b). 
(b-a)=(3+1)------> 4
= 1/4 ∫ cos(x)/(x² + x + 2) dx (from x=-1 to 3). 

Note that [cos(x)/(x² + x + 2)] does not have an elementary anti-derivative.
 By approximating techniques: 
1/4 ∫ cos(x)/(x^2 + x + 2) dx (from x=-1 to 3) ≈ 0.182951

the answer is
the average value of y = cos(x)/(x² + x + 2) on [-1, 3] is approximately 0.182951