Hint 2

This can be done by constructing the potential as described in the lectures for the case $x>a$. In changing variables from $\theta$ to $r$ you should remember that $r$ is positive definite and so for $x<a$ the lower limit on the $r$ integral is now $a-x$. The potential inside is then

$$\Phi(x)=-{GM\over a},$$

for all $x<a$. (This is consistent with Gauss' Law, which requires that the field inside a spherical shell vanish.)