Is it correct to use the infinity symbol while solving definite integrals, like so:
$$\int_1^\infty \frac{1}{x}\,dx$$ $$=\ln(x)\,\Big|_1^\infty$$ $$=\lim_{b \rightarrow \infty}\left(\ln(b)-\ln(1)\right)$$
or does one have to start putting in the limits in the previous steps, like so?
$$\int_1^\infty \frac{1}{x}\,dx$$ $$=\lim_{b \rightarrow \infty}\int_1^b \frac{1}{x}\,dx$$ $$=\lim_{b \rightarrow \infty}\left(\ln(x)\,\Big|_1^b\right)$$ $$=\lim_{b \rightarrow \infty}\left(\ln(b)-\ln(1)\right)$$
I believe the second longer notation is the "expansion" of what people normally understand by writing the first one. It's always a limit, so you don't have to forget this. It's not a raw substitution. You always operate in the limit way. So both are correct.
Warning: In the second case you assume that you can interchange the limit with the integral, which is not always possible. So be careful with that.