Solve Complicated Integral: 4+ 8/πx +O(x²) at x→0

[wel]

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.

 

[CAF123]

Notice that you are only to consider the case when x is very small, tending to zero. This means you can make a suitable expansion of the exponential function, leaving a much simpler integral.

 

[Saitama]

Notice that you are only to consider the case when x is very small, tending to zero. This means you can make a suitable expansion of the exponential function, leaving a much simpler integral.

Or maybe do this:
$$I=I(0)+I'(0)x+O(x^2)$$