# Laplace's method for integration

I am trying to compute $$I(\lambda) = \int_{0}^1 \frac{x}{\sqrt{1+x^4}}e^{\lambda x}\mathrm{d}x$$ for large, real, positive $$\lambda$$. I'm attempting this with Laplace's method as suggested, however I fail to see how this would work considering $$f(x) = x$$ has no maximum on the interior of the interval $$(0,1)$$ and $$f''(x) = 0$$, which would be problematic in the formula for Laplace's method. Am I missing a small substitution trick here, or is the question poorly written?

• The main contribution comes from $x \lesssim 1$. Change $x \mapsto 1 - x$. – Felix Marin Oct 22 '20 at 23:45

• Thanks! I split the final integral into 2 parts and used Watson's lemma on both and got the next term to be $-\frac{\sqrt{2}}{2}\frac{e^\lambda}{\lambda^{-3}}$ – logistress Oct 23 '20 at 12:08