It's the part before and after "Thus".
$I = \ldots = \int e^{ax} \cos bx \ \mathrm{d}x = \frac{1}{b} e^{ax} \sin bx + \frac{a}{b^{2}} \cos bx - \frac{a^{2}}{b^{2}} I.$
Thus
$\left( 1 + \frac{a^{2}}{b^{2}} \right) I = \frac{1}{b} e^{ax} \sin bx + \frac{a}{b^{2}} \cos bx + C_1,$
and
$\int e^{ax} \cos bx \ \mathrm{d}x = I = \frac{be^{ax}\sin bx+ae^{ax} \cos bx}{b^2+a^2}+C.$
Were does the "+1" come from? I thought this was an old "move to other side of equal-sign" until that +1 spawned in my face hehe.