Why would you parameterize in terms of 't' and then fail to use 't' in your limits of integration?

It seems to me that your off-handed discarding of the parameterization ("not even needed") and your casual use of 'x' for limits may be misguided.

Try t in [0,4] You may notice that this is [ln(e^0),ln(e^4)]. If you examine the relationship between x and t, this should be the relationship.