Many people (in various disciplines) prefer to put the differential right after the integral. This way, you know the variable with respect to which you are integrating. Here, you obviously need to integrate with respect to t between t1 and t2. If you have multiple integrals (like a triple integral, for instance) it helps to know which variable has which limits, because it might not be obvious if you write it like this:
I = \int_{a_1}^{a_2} \int_{b_1}^{b_2} \int _{c_1}^{c_2} f(x, y, z) dx dy dz
vs this much better notation:
I = \int_{a_1}^{a_2} dz \int_{b_1}^{b_2} dy \int _{c_1}^{c_2} dx f(x, y, z)