[Insert paragraph about PreCalculus here.]
Theoretically, integrated math 1-3 should cover everything that you need to know in order to then move straight into Calculus 1. However, it is very rare for these classes to actually cover everything needed for Calculus. This is simply because there is so much material to cover in each course, that it is impossible to spend enough time on each topic to cover everything thoroughly. You get a taste of everything that you'll need for Calculus, but in order to be as prepared as possible going into a Calculus class, you'll likely need the in-depth practice that a PreCalculus or College Algebra course will provide.
In some schools, Math 1, Math 2, and Math 3 are specifically planned to prepare you for PreCalculus, while Honors Math 1, Honors Math 2, and Honors Math 3 are meant to prepare you to go straight into Calculus. However, even if you've taken the honors courses, in some schools the teachers will still recommend that you take PreCalculus after Math 3, just to make sure you're really solid on everything before moving on to Calculus 1.
Another thing to consider if you are thinking about taking a Calculus class is, would you be taking an in-depth, college-level calculus class, in order to receive college credit (such as an AP or Concurrent Enrollment class)? Or are you taking a not-quite-so-rigorous, basic introduction to Calculus? If you are taking a college-level course, then it will likely be assumed that you know all the algebra concepts you should have learned by now. The professor will not slow down or stop to review factoring or other algebra details, because everyone should already know those things by now. There's a lot to cover, and no time to pause the class and take a few days to review. On the other hand, if you are taking more of a basic introductory, conceptual-based Calculus class, that isn't as fast-paced, then you probably won't go as in-depth with the detailed complex calculations, and when you do have to use more advanced algebra techniques, your instructor will be more likely to take the time to review them as needed.
Long story short: ask the math teacher/teachers at your school. Chances are, if you've completed Math 3 or even honors Math 3, you should still probably take a PreCalculus or College Algebra course before you start Calculus.