I ran across this integration problem that has an interesting pattern.
$\int_{0}^{(n-1)\pi}\frac{1}{\tan^{n}(x)+1}dx=\frac{(n-1)\pi}{2}$
I evaluated increasing values of n up to n=10, and the result is always one half the upper limit of integration.
Why is this?. That is, how could we show it. I graphed it and as n gets larger, the graph resembles rectangles of equal size stacked up along the x-axis.
Thanks.