Hello I am struggling with the following integral:

$$ I(t) = \displaystyle\int\limits\_{0}^{\infty} e^{jx(t+j)}dx $$

Where $j = \sqrt{-1}$. I think the answer should be:

$$ I(t) = \frac{1}{1-jt} $$

However I don't understand how to evaluate the limit in the following:

$$ I(t) = \frac{1}{j(t+j)} \; \big( \underset{x \rightarrow \infty}{\lim} e^{jx(t+j)} -1 \big)$$

I tried expanding it using the Euler formula, but it doesn't help because the cosine doesn't have a limit, right? Any help?