Asymptotic expansion / analysis of this integral

As $$M \to +\infty$$, how could I make a good asymptotic analysis of this integral?

$$\int_0^1 \dfrac{\cos(M x)}{1 + x^2} e^{-M (x^2 - 1/9)}\ \text{d}x$$

The exponential term shall dominate, yet I have no clue in who to deal with $$\cos(Mx)$$. I tried to apply geometric series for $$\frac{1}{1+x^2}$$ but an infinite sums might not be the best deal.

I am not sure if/when to use Taylor series. I thought about Laplace Method, but I think it could not work because of the $$\cos(MX)$$ function... Any hint?

Thank you!

• Didn't I just see this question on MSE, and it had an answer in a comment? The problem there was no effort shown. Here, they're gonna shut this down in minutes. – skbmoore Sep 15 at 20:27
• @skbmoore I have already done some effort by reducing the initial integral to that one, which might not be much but still it's useful. What I tried, I wrote above with no result. Or no meaningful reason. It will for sure be shut down in minutes if you downvote instead of taking this question as interesting and spread it as it (because it is interesting, and the fact that you cannot provide an answer does not mean you have to downvote). – Turing Sep 15 at 20:45
• I don't think you understand the quality of questions this site seeks: research level. Your problem can probably be solved with standard techniques out of textbooks. It was an appropriate question for MSE. Try there again, but with more effort. – skbmoore Sep 15 at 20:59

For large $$M$$ the integrand contributes in the range $$x\lesssim 1/\sqrt M$$, so we can neglect the denominator $$1+x^2$$. The integral then has a closed form expression, $$\int_0^1 \cos (M x) e^{-M (x^2 - 1/9)}\ \text{d}x=\tfrac{1}{4}\sqrt{\pi }M^{-1/2}e^{M/9} e^{-M/4} \left[\text{erf}\left(\left(1+\tfrac{i}{2}\right) \sqrt{M}\right)-i \,\text{erfi}\left(\left(\tfrac{1}{2}+i\right) \sqrt{M}\right)\right].$$ The expression in square brackets tends to 2 in the limit $$M\rightarrow\infty$$, leaving us with the asymptotics $$\int_0^1 \dfrac{\cos M x}{1 + x^2} e^{-M (x^2 - 1/9)}\ \text{d}x\rightarrow \tfrac{1}{2}\sqrt{\pi }M^{-1/2}e^{M/9} e^{-M/4}.$$