I am definitely a fan of memorizing the grammar of a subject. For calculus, what would that be? Basic derivative and integral facts. Of the formulas you have written there, you definitely need to know your sine and cosine. I would say all the others are optional. You can derive them all (pun intended) by knowing how to differentiate all the basic trig functions: just turn each differentiation rule into an integral rule.
That's how I would proceed. Make sense?