Is it always true that integral of nonnegative function is non negative

Question

If $f(x)\geq 0$, is it true that $\int f(x)dx \geq 0$?

Answer 1

To clarify, $\int f(x) dx$ denotes the primitive of $f$. That is, any function $F(x)$ such that $F'(x)=f(x)$. There are infinitely many such functions for any given $f$, since you may add any constant to one primitive to get another, and the constant you add can be negative and have arbitrarily large magnitude, so there exist infinitely many primitives of a given $f$ that are negative at some point. If you are asking about $\int_a^bf(x)dx$, then if $b\geq a$ and $f(x)\geq0$ for all $x\in [a,b]$ then $\int_a^bf(x) dx\geq0$ and $\int_b^a f(x)dx\leq 0$.