Finding Mean Angular Velocity of Rotating Body

[paragchitnis]

Homework Statement

A solid body rotates with deceleration about a stationary axis with an angular deceleration
$$\beta$$ $$\propto$$ $$\sqrt{\varpi}$$ , where $$\varpi$$ is the angular velocity. Find the mean angular velocity of the body averaged over the whole time of the rotation if at the initial moment of time the angular velocity was $$\varpi$$₀

Homework Equations

$$\varpi$$f = $$\varpi$$i + $$\beta$$t
$$\varpi$$_avg = $$\varphi$$/t

The Attempt at a Solution

The given answer is <$$\varpi$$> = $$\varpi$$₀/3

 

[member 553083]

, but I don't understand how to get there. We know that \varpif = \varpii + \betat, so we can integrate the expression \int_0^t (\varpii + \betat) dt. But then what?Can someone explain to me how to solve this?