If a satellite of mass $m$ is orbiting a planet of mass $M$ with radius $r_1$ and orbital speed $v_1$ and is brought to orbit at $r_2$ with speed $v_2$, its kinetic energy changes by a quantity
$$ \Delta E_k = \frac{1}{2} GMm \left(\frac{1}{r_2}-\frac{1}{r_1}\right)$$
Since the total energy in orbit 2 is equal to the total energy in orbit 1 plus the work done to change orbit:
$$E_{tot}^{(2)}=E_{tot}^{(1)}+W_{12}$$ $$\frac{1}{2}mv_1^2 - \frac{GMm}{r_1} = \frac{1}{2}mv_2^2 - \frac{GMm}{r_2} + W_{12}$$
I obtain that the work done to change orbit is
$$W_{12}= \Delta E_k - \Delta E_p = -\frac{1}{2} GMm \left(\frac{1}{r_2}-\frac{1}{r_1}\right)$$
I thought I could calculate this assuming that the minimum work required to change orbit is that done along a radius (either against or with gravity) by a force equal to gravity (in magnitude). Using the definition of work done:
$$W_{12} = \intop_{r_1}^{r_2} \frac{GMm}{r^2} dr$$
I don't see how to obtain the factor $1/2$. What am I doing wrong?