Question

Let f be the function given by f(x)=3ln⁡(2+x2)cos⁡x. What is the average value of f on the closed interval 2≤x≤6?

Answer 1

The average value of f(x) over [2, 6] is given by the definite integral,

$\displaystyle f_{\rm ave[2,6]} = \frac1{6-2} \int_2^6 3\ln(2+x^3)\cos(x) \, dx$

and is approximately -1.67284.