Consider the integral

\begin{equation}

I(x)=\int^{2}_{0} (1+t) e^{xcos[\pi (t-1)/2]} dt

\end{equation}

show that

\begin{equation}

I(x)= 4+ \frac{8}{\pi}x +O(x^{2})

\end{equation}

as $x\rightarrow0$.

=> Using integration by parts, but its too complicated for me because of huge exponential term.

please help me.