Let $X\rightarrow \mathbb P^n_{\mathbb C}$ be a double cover ramified over a smooth hypersurface $B$ of degre $2d$. In the case of hypersurfaces of $\mathbb P^n$ one can determine the integral cohomology using Lefschetz hyperlane section theorem and universal cefficient.

Q. Are there some simple technics allowing to compute $H^k(X,\mathbb Z)$ for any $k$?