  We prove some Lambert series which were stated by Gosper without proof or
reference. As an application, we shall evaluate integrals involving Ramanujan
theta function $\psi(q)$. Furthermore, motivated by Ramanujan's identities for
$q\psi^4(q^2)$ and $\fr{\psi^3(q)}{\psi(q^3)}$, we shall evaluate the squares
of $q\psi^4(q^2)$ and $\fr{\psi^3(q)}{\psi(q^3)}$ in terms of Lambert series.
