# What is the average value of the function u(x) = 10xsin(x^2) on the interval [0,sqrt pi]?

Dec 20, 2016

See below.

#### Explanation:

The average value is

$\frac{1}{\sqrt{\pi} - 0} {\int}_{0}^{\sqrt{\pi}} 10 x \sin \left({x}^{2}\right) \mathrm{dx} = \frac{5}{\sqrt{\pi}} {\int}_{0}^{\sqrt{\pi}} 2 x \sin \left({x}^{2}\right) \mathrm{dx}$

$= \frac{5}{\sqrt{\pi}} {\left[- \cos \left({x}^{2}\right)\right]}_{0}^{\sqrt{\pi}}$

$= \frac{12}{\sqrt{\pi}}$

Pedantic Note

$\frac{12 \sqrt{\pi}}{\pi}$ does NOT have a rational denominator.