  We develop efficient numerical integration methods for computing an integral
whose integrand is a product of a smooth function and the Gaussian function
with a small standard deviation. Traditional numerical integration methods
applied to the integral normally lead to poor accuracy due to the rapid change
in high order derivatives of its integrand when the standard deviation is
small. The proposed quadrature schemes are based on graded meshes designed
according to the standard deviation so that the quadrature errors on the
resulting subintervals are approximately equal. The integral in each
subinterval is then computed by considering the Gaussian function as a weight
function and interpolating the smooth factor of the integrand at the Chebyshev
points of the first kind. For a finite order differentiable factor, we design a
quadrature scheme having accuracy of a polynomial order and for an infinitely
differentiable factor of the integrand, we design a quadrature scheme having
accuracy of an exponential order. Numerical results are presented to confirm
the accuracy of these proposed quadrature schemes.
