I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is the Fourier transform of the density function of $x = \ln(S)$ (under measure $Q$), then the probability of exercise under $Q$ (that, is probability $x > \ln(K)$) is
$$F_j (x(T) > a) = \frac{1}{2\pi} \int _{\mathbb{R}} f_j(\phi) \Bigg(\int_a^{\infty} e^{-i\phi x} \mathrm{d}x\Bigg) \mathrm{d}\phi\text{.}\tag{1}$$
This equation makes sense to me. The author then says, equation (2.17),
A further straightforward calculation yields
$$F_j (x(T) > a) = \frac{1}{2} + \frac{1}{2\pi}\int_{\mathbb{R}}f_j(\phi)\frac{e^{-i\phi a}}{i\phi}\mathrm{d}\phi \text{.}\tag{2}$$
This equation does not make sense to me. Where did equation (2) come from? In particular, where did the $\frac{1}{2}$ come from?