I have a question concerning Mermin's 1967 paper "Existence of Zero Sound in a Fermi Liquid". The condition on zero sound is given by the equation
$$\lambda_n>\eta^{-1}\int \frac{d\hat{n}}{4\pi}|\chi(\hat{n})|^2\cos\theta+\int\frac{d\hat{n}}{4\pi}\int\frac{d\hat{n}'}{4\pi}\chi^*(\hat{n})B(\hat{n}\cdot \hat{n}')\chi(\hat{n}') $$
where $\chi(n)$ is an arbitrary function, $\eta v_F$ is the phase velocity of the mode, $\lambda_n$ is an eigenvalue, and $B(\hat{n}\cdot \hat{n}')$ is the spin-symmetric forward scattering amplitude. By proposing the trial function
$$\chi(\hat{n})=\chi(\theta,\,\phi)=\begin{cases}\frac{A}{\eta-\cos\theta},\quad &0<\theta<\theta_0 \\ 0,\quad &\theta_0<\theta<\pi \end{cases} $$
where $A$ is a normalization constant and we take the assumption that $B(x)>B_{\theta_0}>0$. The above condition becomes (in a simplified form)
(1) $$\lambda_n>1-\eta^{-1}A^2 \int_{\cos\theta_0}^1 \frac{dx}{\eta-x}+\frac{A^2}{2}B_{\theta_0}\int_{\cos\theta_0}^1 \frac{dx}{\eta-x}\int_{\cos\theta_0}^1 \frac{dx}{\eta-x}$$
I understand how he got the last term, but the first term I find confusing. Directly putting in his trial function, I get
(2) $$\eta^{-1}\int \frac{d\hat{n}}{4\pi}|\chi(\hat{n})|^2\cos\theta=\frac{\eta^{-1}}{2}A^2\int d\theta \sin\theta \cos\theta \frac{1}{(\eta-\cos\theta)^2}\equiv \frac{\eta^{-1}}{2}A^2\int dx \frac{x}{(\eta-x)^2}$$
and I really don't see how (2) reduces to the first two terms in (1). Was some approximation utilized that wasn't mentioned? Any clarification would be appreciated.
