How would I go about setting up a Biot and Savart Integral using Vectors?

Here is an exercise we had in class:

I tried to set it up using Vectors, and figured dl to be <-L,L> and r to be <½L, -½L>. After the cross product, I am not sure how to handle the integration, since r can vary. Would it be a double integral, or am I overthinking it?

The instructor gave me the answer, and I have been trying to figure out a way to use Vectors to solve it, since I am fairly comfortable with them. Any help would be appreciated!