I think that there are two ways to tackle this problem by exploiting the analyticity of the integrand . We can calculate it in the real domain by Taylor series or in the complex domain by Cauchy's integral formula.
Solution by Taylor Series
First, let us introduce the function
so we have the equation
. By Taylor expansion, we have
It is easy to show that all the coefficients of this Taylor expansion vanish except for the first one. Namely, . The -th derivatives of at must be in the form of
Hence, we have .
Solution by Cauchy's Integral Formula
First, let us introduce the same function
The integrand can be converted to a complex exponential,
Cauchy's Integral formula at is
If we take , and , we have
Note that we have