So we have:

Using a simple u-substitution, we get

We have a case where both of our bounds will lead to an improper integral, so its best to split this up (for your own sanity, although you could just as easily keep it the same since its not discontinuous over the reals). It's arbitrary where we split the integral, but - as you see - its best to choose 0 since it'll tide up nicely so:

Remember the graph of the exponential function, as it approaches negative negative values of "x" it approaches zero, and as it apporaches positive values of "x" it approaches infinity. In one of our limits, we have "t" approaching negative infinity: a negative times a negative is a positive; thus the exponential is being raised to "positive infinity". In the other, our exponential is being raised to a negative, thus, as it gets larger negative, the values approach zero. And so we have: