You should be able to immediately integrate [MATH]e^{-\frac{t}{T}}[/MATH]. That doesn't require parts, just a simple substitution, if that.
What you might want to do is first just to check the answer they give; in fact, that is really enough to answer the question. Differentiate the given function [MATH]\phi(t)[/MATH], and show that (a) the derivative is what you were given, and (b) [MATH]\phi(0) = 0[/MATH].