I'm working on Deriving the characteristic function for the maximum of a drifted brownian motion $W(t) + at$, I've found a paper deriving what I need, but I don't understand the following step they make (without explanation):
$$ \begin{aligned} g(y) &= -\sqrt{\frac{2}{\pi}}\frac{1}{T^{3/2}}e^{2ya}\bigg[\int_{-\infty}^y[x-(2y+Ta)]e^{-[x-(2y+Ta)]^2/2T}dx + Ta\int_{-\infty}^ye^{-[x-(2y+Ta)]^2/2T}dx\bigg]\\ &=\sqrt{\frac{2}{T\pi}}e^{2ya}\bigg[e^{-(y+Ta)^2/2T} - a\sqrt{2\pi T}\,\Phi\left(\frac{-y-Ta}{\sqrt{T}}\right)\bigg] \end{aligned} $$ where $\Phi$ is the cdf for the standard normal distribution $$ \Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2}dt. $$ I just don't know where to start with this monster. I tried rewriting the second expression in the final expression but I don't know how to continue from: $$ \Phi\left(\frac{-y-Ta}{\sqrt{T}}\right) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\big(\frac{-y-Ta}{\sqrt{T}}\big)}e^{-t^2/2}dt $$ Any help at all with this would be appreciated!!