TITLE: Find the average value of the function...
QUESTION [5 upvotes]: Find the average value of the function $$F(x) = \int_x^1 \sin(t^2) \, dt$$ on $[0,1]$.

I know the average value of a function $f(x)$ on $[a,b]$ is $f_\text{avg}=\dfrac1{b-a} \int_a^b f(x) \, dx$, but I don't know how to apply that to this question... The function loos like the Fresnel integral? But that doesn't quite help me either.

REPLY [7 votes]: $$
\begin{align}
\int_0^1\int_x^1\sin\left(t^2\right)\,\mathrm{d}t\,\mathrm{d}x
&=\int_0^1\int_0^t\sin\left(t^2\right)\,\mathrm{d}x\,\mathrm{d}t\\
&=\int_0^1\sin\left(t^2\right)\,t\,\mathrm{d}t\\
&=\frac12\int_0^1\sin\left(t^2\right)\,\mathrm{d}t^2\\
&=\frac12\left[-\cos\left(t^2\right)\right]_0^1\\[3pt]
&=\frac{1-\cos(1)}2
\end{align}
$$