Is a definite integral of $f(x)$ rational if $f(x)$ is rational at the endpoints and f(0)=0?

Question

It seems that in many situations, the fact that $f(x)$ is rational at $x=a,b$ and $f(0)=0$ and $f$ is integrable is enough to say that $$\int_a^bf(x)dx$$ is rational. Is this true? If not, can you give a pretty-large class of commonly discussed combinations of elementary functions for which it is true?

(E.g. It's true for rational-coefficient polynomials, but is it true for polynomials in $ln(x)$?)

Answer 1

Take the function $x^2$ on $[0,\sqrt{2}]$

Answer 2

I can give an easy counterexample: $$\int_0^{\tfrac{\pi}6}\sin x\,\mathrm dx=\cos x\biggm|_{\tfrac\pi 6}^0=1-\frac{\sqrt 3}2, $$ yet $\sin 0=0$, $\sin\frac\pi 6=\frac 12$.