Your t can be negative, because the anti derivative of 1/t is ln|t| + C. The variable is an absolute value, which of course means that if it is negative you just make it positive
I am not entirely sure how to solve this problem because you did not give an x component to go with the t. Usually you are given x, or are can find it out. The ln|-1| is 0, so that would mean C would equal whatever x value you find. Hope that explains it