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I am stuck in integrating the following integral $I(x) = \int_{a}^{b}t^{2}e^{-xf(t)}\,dt$ where $f(t)=(1-e^{-t})^2$. The main problem here is at the point where $f(t)$ is zero, $t^2$ is zero. I am not sure how to evaluate this integral. Thank you very much in advance.

Vip
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    You will need higher order terms from the Laplace approximation. See the last section here: https://www.math.unl.edu/~scohn1/8423/intasym1.pdf – Gary Apr 20 '20 at 09:58
  • Thanks. I will check the file. – Vip Apr 20 '20 at 10:28
  • To make sure. 0 is interior to the bounds a and b? – AHusain Apr 20 '20 at 10:54
  • No. In fact I need to integrate from zero to an arbitrary upper limit – Vip Apr 20 '20 at 10:55
  • You can have a problem because $f(t)$ tends to a finite limit as $t\to +\infty$. You can only look at finite intervals and for larger upper limits of integration you will need larger and larger values of $x$ to keep the contribution of the tail small. – Gary Apr 20 '20 at 14:13

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