I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed that $B=0$. I wonder why it is true, because, in my opinion, there should be an extra term coming from the integration by parts: $$\begin{align}B&=\theta\int_0^1 w(y)\overline{w}'(y)dy\\ &=\theta\int_0^1 e^{2Ry}[R(1+\lambda y)^2+\lambda(1+\lambda y)]dy\\ &=\theta\left(\int_0^1 e^{2Ry}R(1+\lambda y)^2dy+\frac{1}{2}\int_0^1e^{2Ry}d((1+\lambda y)^2)\right)\\ \\ &=\theta \Bigg[\frac{1}{2}e^{2Ry}(1+\lambda y)^2\Bigg]_{y=0}^{y=1},\end{align}$$ where the last equality follows from doing integration by parts. I think this paper have been read and checked by many people. I would appreciate if I can get help from some of you who are familiar with this paper. Thank you very much.

P.S. I have posted the question on Mathematics Stack Exchange few days ago. But I did not get an answer. I am sorry if this question does not appear to be about research level mathematics, and I will delete my question if it is the case.