I'm asked to explain why the following integral is improper and determine whether the integral is convergent or divergent. I really am not sure how to do these problems and I am unsure on where to start with this. The integral is:
$\int_0^\frac {\pi}{2} sec(x) dx$
Can someone explain to me how I approach these sort of problems?
Thanks
Best Answer
A definite integral is improper if either the interval you're integrating over is infinite (for example: $\int_1^\infty \frac{1}{x^2} dx$) or if at least one point in the interval you're integrating over is not in the domain of the integrand. In this case, the latter is true; there's a number $c$ in $\lbrack 0, \frac{\pi}{2}\rbrack$ at which $\sec x = 1/\cos x$ is undefined. Do you see where it is? Conveniently there's only one such $c$ in this exercise.
Once you know where $c$ is, the improper integral is defined to be $$ \lim_{t \to c^-}\int_0^c \sec(x) dx + \lim_{t\to c^+}\int_c^{\pi/2} \sec x dx. $$ if $0 < c < \pi/2$, $$ \lim_{t\to 0^+}\int_c^{\pi/2} \sec x dx $$ if $c = 0$, and $$ \lim_{t \to (\pi/2)^-}\int_0^c \sec(x) dx $$ if $c = \pi/2$, where there's a different definition if $c$ equals one of the end points, since otherwise one of the integrals in the first sum doesn't make sense.