Laplace's Method for Integration of the Integral I(x)

[wel]

Consider the integral
\begin{equation}
I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt
\end{equation}
Use Laplace's Method to show that
\begin{equation}
I(x) \sim \frac{4\sqrt{2}e^{x}}{\sqrt{\pi x}} \end{equation}
as $x\rightarrow\infty$.

=> I have tried using the expansion of $I(x)$ in McLaurin series but did not get the answer.
here,
\begin{equation}
h(t)=cos(\frac{\pi(t-1)}{2})
\end{equation}
$h(0)= 0$

$h'(0)= \frac {\pi}{2}$

Also $f(t)= (1+t) \approx f(0) =1$, so that

\begin{equation}
I(x)\sim \int^{\delta}_{0} e^{x \frac{\pi}{2}t} dt
\end{equation}

after that I tried doing integration by substitution $\tau = x \frac{\pi}{2} t$ but did not get the answer.

please help me.

 

[Simon Bridge]

Laplaces method?
http://en.wikipedia.org/wiki/Laplace's_method
... see "other formulations" and compare with what you did.
Did you correctly identify the function h(t)?

Please show your best attempt (working and reasoning) using this method.
I have a strong feeling about where you went wrong but I don't want to waste my time and yours on a guess.