$f(x) = \int\limits_0^x {{\pi ^t}(t - e)(t - \pi )dt} $

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 $( - \infty ,e] \cup [\pi ,\infty )$.