and exactly why would you factor a function of t out of a dt-integral? you can only factor out constants (with respect to the variable of integration) remember?

Hint: you know the power series for the given log function, just write that in. then divide it through by t. then integrate term by term. well, that's not really a hint, that's how to do the problem

by the way, your power series for the log is incorrect, you wrote the power series for 1/(1 - t) ...which is NOT the integral of ln(1 - t), by the way