Find interval(s) in which f increases.
Using the formula $\frac{d}{{dx}}\left[ {\int\limits_a^x {g(x,u)du} } \right] = g(x,x) + \int\limits_a^x {\frac{\partial }{{\partial x}}g(x,u)du} $, we have
$f'(x) = {\pi ^x}(x - e)(x - \pi )$
Let us look at the sign scheme for f'(x).
For increasing function, we have $f'(x) \ge 0$
So, f increases in the intervals $( - \infty ,e] \cup [\pi ,\infty )$.