Let $\Delta(x)$ denote the error term in the Dirichlet
divisor problem. Our main results are the asymptotic formulas
for the integral of the cube and the fourth power of
$\Delta(x)$. The exponents that we obtain in the error terms,
namely $\beta = {\sfrac{7}{5}}$ and $\gamma =
{\sfrac{23}{12}}$, respectively, are new. They improve on the
values $\beta = {\sfrac{47}{28}}, \gamma = {\sfrac{45}{23}}$,
due to K.-M. Tsang. A result on integrals of $\Delta^3(x)$
and $\Delta^4(x)$ in short intervals is also proved.