Notation issue with the integration of exponents.

[Fallen Seraph]

I'll not go into the details of the full question, because they are irrelevant to my problem.
Basically I have to integrate
$$\int_{0}^{\infty} exp (\iota\omega-\alpha)t dt$$

Which is a nice and easy integration, but it's putting in the limits that bothers me.

I simply wrote the exponent as $$((\iota \omega - \alpha)t)$$ because I didn't feel like writing an extra minus sign. I see no reason why I could not have written it

$$(-( \alpha -\iota \omega )t)$$

Which gives a finite answer when putting in the limits, whereas the first way of writing it gives an infinite answer. Could someone explain why one of these notations are incorrect?

 

[Shooting Star]

Integral[0, inf] exp(at)dt converges, i.e., has a finite value only when a<0.