Average value of $f(x)=\int_x^1 \cos(t^2) \,dt$ on the interval $[0,1]$. Find the average value of the function 

$$f(x)=\int_x^1 \cos(t^2) \,dt
$$ 

on the interval $[0,1]$.

\begin{align}
\overline{f} & =\frac 1 {1-0} \int_0^1 \int_x^1\cos (t^2) \; dt \, dx = \int_0^1 \int_0^t\cos(t^2)\;dx\,dt \\[10pt]
& =\int_0^1t\cos(t^2)\,dt = \left[\frac 1 2 \sin(t^2)\right]_0^1 = \frac 1 2 \sin1
\end{align}
