# Integration

• April 17th 2014, 01:24 PM
grandy
Integration
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.
• April 17th 2014, 03:13 PM
romsek
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.
• April 18th 2014, 04:44 AM
Jester
Re: Integration
Another way

$I(x) = I(0) + I'(0)x + O(x^2)$

Then calculate $I(0)$ and $I'(0)$.