I don't understand how to solve the following integral in order to obtain the reported solution found in the book "Fundamentals of atomic mechanics" written by Enrico Persico.
The following integral ${\Delta}y$ is named as the eigendifferential:
$${\Delta}y\ =\ \int_{\lambda_0}^{\lambda_0+\Delta\lambda} y_{\lambda}d\lambda$$ where the function $y_{\lambda}$ would be either one of the following eigenfunction where $a_{\lambda}$ is constant respect to $x$:
$$y_{\lambda}\ =\ a_{\lambda}e^{i\sqrt{\lambda}x}$$
$$y_{\lambda}\ =\ a_{\lambda}e^{-i\sqrt{\lambda}x}$$
The solution reported in the book is obtained by putting $\lambda=\lambda_0+\epsilon$ and by neglecting the squares of $\epsilon$ which I can't find when I try to solve the integral.
Therefore the solution for the eigendifferential integral (where $\lambda$ has been written instead of $\lambda_0$) is:
$$\Delta y\ =\ a_{\lambda}e^{i\sqrt{\lambda+\frac{\Delta \lambda}{2}}x}\frac{\sin{\frac{x\Delta \lambda}{4 \sqrt{\lambda}}}}{\frac{x \Delta \lambda}{4 \sqrt\lambda}} \Delta\lambda$$
It is also stated that the notation $\Delta y$ is used in order to consider the limit where $\Delta \lambda \to 0$ so the interval of continuous eigenvalues $(\lambda_0 ,\lambda_0+\Delta \lambda)$ will be infinitesimal.
Unfortunately for me, no step by step solution nor hint are given by the author therefore I can't obtain the result, at least not in the form here given. Any hint or help in the solution process would be appreciated. Thank you in advance.