The value of the line integral of |B| around the closed path shown in the figure is 3.77 x 10^-6 T m. What is I3?

You must mean the line integral of the vector B, not its length |B|…

Because Ampere’s law in integral form states:

closed-loop-integral of B . dl = mu0 * I_enclosed

Here the contour is traversed in the clockwise direction, so we need to have the net enclosed current going into the page, which is I3 + 6.0A – 4.0A

Hence

4pi*10^-7 T m / A * ( I3 + 2.0A) = 3.77*10^6 T m

I3 + 2.0A = (3.77*10^-6)/(4 pi * 10^-7) A

I3 = 3.0 A