# Complicated integral

1. Apr 17, 2014

### wel

Consider the integral

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

show that

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

as $$x\rightarrow0.$$

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

2. Apr 17, 2014

### 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.

3. Apr 18, 2014

### Pranav-Arora

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

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