Answer
The integral does not exist.
Work Step by Step
$\int_{0}^{\pi} sec^2~x~dx = \int_{0}^{\pi} \frac{1}{cos^2~x}~dx$
The graph of $~~\frac{1}{cos^2~x}~~$ has a vertical asymptote at the point $x=\frac{\pi}{2}$. Therefore, the function is not continuous on the interval $[0,\pi]$ and we can not apply the Fundamental Theorem to calculate the value of this integral.
The integral does not exist.
