1. r(t) = <t^2, t^3> for t:[1,5]

You're going to need to post the entire problem if you want us to know what you're solving for. In general, if you want to take the line integral with respect to a variable (in this case, dx for x or dy for y) along a curve C given by a parametric eq., you have to express the integral in terms of the parameter (t in this case).

integral(f(x,y))dx = integral(f(x(t),y(t))*dx/dt)dt for upper and lower t.

2. You want to compute the line integral of a function f(x(t),y(t)) along a curve C give parametrically with respect to a variable (either x or y). There should be a formula for this (I mentioned it above for the case where you're computing the line integral with respect to x). Convert the integral with respect to a variable to the integral with respect to the parameter, t.