Rational integral conjecture of $(x^{2k} - \sin(\pi x))^2$

Question

I was messing around, and came up with the following conjecture:

$$\int_{-1}^1(x^{2k} - \sin(\pi x))^2dx = \frac{4k+3}{4k+1}$$

More generally, it seems that for any polynomial $P$ with rational coefficients we have that $\int_{-1}^1(P(x^2) - \sin(\pi x))^2dx$ is rational. Which leads me to ask three questions:

1. Is the conjecture correct?

2. Is the more general conjecture about the rationality of $\int_{-1}^1(P(x^2) - \sin(\pi x))^2dx$ correct?

3. If yes, is there a formula that given $P$ (e.g. as a series of coefficients $a_k$) computes $\int_{-1}^1(P(x^2) - \sin(\pi x))^2dx$?