When is the closed subscheme of $\Bbb P^n_{\Bbb C}$ cut out by a homogeneous polynomial $f$ integral?

Question

I am new to algebraic geometry. I am trying to understand whether closed subschemes of $\mathbb P^n_{\mathbb C}$ is an integral scheme.

Let $X = \operatorname{Proj} (A/(f))$ where $f$ is an homogeneous irreducible element of $A=\mathbb C[x_0,\ldots,x_n]$. I am trying to understand whether $X$ is an integral scheme.

It is clear to me that $X$ is a reduced scheme. It would be helpful, if someone can give me hints regarding irreducibility part.