Can anyone see the equality in the red squere is true? It is taken from the supplementary material for this paper.
I am not sure if it is relevant but here, $\phi(x)$ represents the density function (PDF) of normal distribution and $\Phi(x)$ denotes the cumulative normal distribution function (CDF). i.e. $$\Phi(x) = \int_{-\infty}^x \phi(t)dt$$
Thanks!
It´s integration by parts:
In $\int_a^b v^{'}udx=[uv]_a^b-\int_a^bvu^{'}dx$ put:
$v^{'}(x)=\phi(x)\Phi(\alpha x)$ and $u(x)=\int_{-\infty}^x\phi(y)\Phi(\alpha y)dy$ and $a=-\infty,b=\infty$ and use the fundamental theorem of calculus.