From the definition of dr

Take the magnitude of dr

You can find the values of in terms of t easily. Then get |dr| on its own and integrate it between t=0 and t=1

Edit. Didnt see that the density wasn't constant.

After you get an expression for dr in the form |dr|=g(t)dt since the density is 1+t find the integral of (1+t)g(t)dt between 0 and 1