I'm read this paper called "Closed orbits and the regular bound spectrum'' by Berry and Tabor, and they use partial integration to cancel terms in the expression (24) (paragraph under (28) explains this). They use uniform approximation to find an integral. They state that they use partial integration to find $$\int_{a/\sqrt{\hbar}}^{b/\sqrt{\hbar}}\mathrm{e}^{-\frac{i\beta}{2}X^2}dX$$ I use different notation here to simplify the writing.
I have two questions: 1) When they say partial integration do they mean integration by parts? I presumed so.
2) I don't know how integration by parts would solve this: Integration by parts is the following $$uv|^b_a-\int_a^bu'vdx=\int_a^buv'dx$$ picking $v'=1$ and $u=\mathrm{e}^{-\frac{i\beta}{2}X^2}$ we have $$\int_{a/\sqrt{\hbar}}^{b/\sqrt{\hbar}}\mathrm{e}^{-\frac{i\beta}{2}X^2}dX=\mathrm{e}^{-\frac{i\beta}{2}X^2}x|^{\frac{b}{\sqrt{\hbar}}}_{\frac{a}{\sqrt{\hbar}}}-\int^{\frac{b}{\sqrt{\hbar}}}_{\frac{a}{\sqrt{\hbar}}}i\beta X^2\mathrm{e}^{-\frac{i\beta}{2}X^2}dX$$ this isn't helping...
EDIT: I had a chat to my supervisor and after a while figured it out. Term III of (24) in the paper has the form $$\left[\frac{\mathrm{e}^{\frac{i\beta X^2}{2}}}{X}\right]_{\Lambda_1/\sqrt{\hbar}}^{\Lambda_2/\sqrt{\hbar}}.$$ Looking at the anti derivative, we notice it splits into two terms, the form the integral has and a term which has a $\hbar$ dependence and vanishes in the semiclassical limit.