is the integral of zero equal to C(a constant) or lnC?

Answer 1

C, generally. Think of the power rule on 0, which is a constant. The integral of any constant k is kx+C. But k=0, so the integral is just C.

However, ln(C) is also a constant. So it doesn't actually matter, so long as ln(C) assumes C>0, otherwise ln(C) is undefined. But you'd want to simplify ln(C) into a new constant, call it C2 = ln(C) anyway. So go with C.

Answer 2

The integral is a constant. Both C and ln C are constants, but conventionally we would say integ(0 dx) = C.

Answer 3

Indefinite integral of 0 = C Definite integral of 0 with upper bound b and lower bound a = F(b) - F(a) = C - C = 0