  Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our
main results are the asymptotic formulas $$ \int_1^X \Delta^3(x){\rm d}x =
BX^{7/4} + O_\epsilon(X^{\beta+\epsilon}) \qquad(B > 0) $$ and $$ \int_1^X
\Delta^4(x){\rm d}x = CX^2 + O_\epsilon(X^{\gamma+\epsilon}) \qquad(C > 0) $$
with $\beta = 7/5, \gamma = 23/12$. This improves on the values $\beta = 47/28,
\gamma = 45/23$, due to K.-M. Tsang. A result on the integrals of $\Delta^3(x)$
and $\Delta^4(x)$ in short intervals is also proved.
