Limit under the sign of integral

Question

I'm studying the following integral \begin{equation*} \int_0^t \tau \cos(c+b\tau+a\tau^2)\text{ d}\tau \end{equation*} in particular, its limit \begin{equation*} \lim_{a\to 0}\int_0^t \tau \cos(c+b\tau+a\tau^2)\text{ d}\tau \end{equation*} and I'm not able to show that it is equal to \begin{equation*} \int_0^t \tau \cos(c+b\tau)\text{ d}\tau \end{equation*} Hence, at this point, I'm wondering if the following passages are true \begin{equation*}\begin{aligned} \lim_{a\to 0} \int_0^t \tau \cos(c+b\tau+a\tau^2)\text{ d}\tau &= \int_0^t \lim_{a\to 0}\tau \cos(c+b\tau+a\tau^2)\text{ d}\tau\\ &=\int_0^t \tau \cos\left[\lim_{a\to 0}(c+b\tau+a\tau^2)\right]\text{ d}\tau\\ &=\int_0^t \tau \cos(c+b\tau)\text{ d}\tau\\ \end{aligned}\end{equation*} I remember (not so well) that there are results, for example the dominated convergence theorem, that say when is possible to switch the order of the limit and integration operation. However, I'm not able to justify the equivalence \begin{equation*} \lim_{a\to 0} \int_0^t \tau \cos(c+b\tau+a\tau^2)\text{ d}\tau = \int_0^t \lim_{a\to 0}\tau \cos(c+b\tau+a\tau^2)\text{ d}\tau \end{equation*}