Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by:
$$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the following statements is TRUE?
(A) $$ is a harmonic polynomial
(B) $$ is a polynomial but not harmonic
(C) $$ is harmonic but not a polynomial
(D) $$ is neither harmonic nor a polynomial
Differentiating under the integral sign I see that $g$ is harmonic. But How do I conclude if it is a polynomial or not??