Is this integral correct?

Question

I used substitution and got that: $$\int_0^\pi \sin x \cdot P_n(\cos x ) \, dx=0$$ where $P_n$ is the $n$-th Legendre polynomial.

Answer 1

Recall that Legendre polynomials are defined as orthogonal polynomials on $[-1,1]$ with weight function $w(x)=1$. In other words, we have by definition $$(P_m,P_n)=\int_{-1}^1P_m(x)P_n(x)dx\sim \delta_{mn}.$$ But, since $P_0(x)=1$, your integral is equal to $(P_0,P_n)$ and therefore it vanishes whenever $n>0$. For $n=0$, however, it is equal to $2$.

Answer 2

If you look at the Power reduction formulas for the cosine, you see that all monomials in your Legendre polynomial involve only cosines of integer multiples of $x$. These are odd functions on the interval $[0,\pi]$ while the sine function is even, thus their product must be odd and the integral is zero.