Ackthually no that is incorrect https://www.masterclass.com/articles/capital-vs-capitol
Capital: The word “capital” can refer to money, uppercase letters, the death penalty, and capital cities that house a seat of government.
Capitol: The word “capitol” always refers to a physical capitol building (such as the US Capitol building) or the area surrounding it (such as Capitol Hill).
For number 4 (or maybe a new number), if a(x) and b(x) are constant, the derivatives d/dx a(x) and d/dx b(x) are zero, so all that is left is that d/dx becomes a partial derivative and can move inside the integral.