Application of the unconscious statistician

A random variable Y is normal with mean u and variance s^2. Given that g(y) =exp(2y) , using a taylor series expansion around u to approximate E[g(y)] and variance(g(y)).

By the unconscious statistician theorem E[g(y)] = integral of exp(2y) * 1/(2*pi*s^2) * exp(-(y-u)^2)/2s^2 dy . How am I suppose to go about using the taylor series to estimate the expected value?