
Integration
Consider the integral
\begin{equation}
I(x)=\int^{2}_{0} (1+t) e^{xcos[\pi (t1)/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.

Re: Integration
Expand the exponential term as a taylor series about x=0.
Keep the terms below 2nd order in t (the first 2 terms)
Plug that into your integral and solve.

Re: Integration
Another way
$\displaystyle I(x) = I(0) + I'(0)x + O(x^2)$
Then calculate $\displaystyle I(0)$ and $\displaystyle I'(0)$.