Is there a way to simplify an integral such as: 

$\lgroup{\int^T_0 r(t)u(t)dt}\rgroup ^2$

The expression appears in an optimal control problem where we seek to minimize the following cost function subject to a control $u(t)$ such that $0 < u(t) < 1 $:

$\lgroup{\int^T_0 r(t)u(t)dt}\rgroup ^2 + \lgroup{\int^T_0 s(t)u(t)dt}\rgroup $

Here $r(t)$ and $s(t)$ are predetermined functions in $t$ and you must choose an admissible trajectory $u(t)$ that will minimize the cost function above.

In theory, one can find such a $u(t)$ using continuous time optimal control, except that the cost function first has to expressed in the form: 

$h(x(T)) + \int^T_0 g(x(t),u(t))\space dt$

This will inevitably involve simplifying the squared integral expression in the cost function.