When people could use taylor's theorem to evaluate the integral and when cannot? When people could use taylor's theorem to evaluate the integral and when cannot? Give one or two counter example please.
There must be something that the taylor's theorem is incapable with otherwise people would had say that this is the golden rule for integration. :)

It works only for differentiable functions. To be more explicit, the function $f$ needs to be $k$-times differentiable (for a $k$-th order expansion) at a point requires differentiability up to order $k−1$ in a neighborhood of said point. See the discussion at http://www.physicsforums.com/showthread.php?t=238794.
$$\int f(x)dx=\\
\sum^\infty_{k=0}\int \dfrac{f^{(k)}(a)}{k!}(x-a)^kdx\text{, only for $k$-times differentiable functions about $x=a$.}$$
For a Taylor series with $k$ terms, $f\in C^k$ - thus, for a Taylor series with $\infty$ terms, $f\in C^\infty$.
For an example where the integral cannot be evaluated, consider about $x=0$ the function $e^{1/x}$.
