Here's the function:
$h(x) = \begin{cases} \cot x & \text{if $x \in [-\pi/2,0)-\{-\pi/3\}$}\\ p & \text{if $x=-\pi/3$}\\ q & \text{if $x=0$} \end{cases}$
Find the value of parameter $p$ (if it exists) so that $h$ is integrable on $\left[-\frac{\pi}{2}, -\frac{\pi}{4}\right]$. Find the value of p for which function h(x) has primitive function on (- pi/2, 0) Find the values of p and q for which h(x) is integrable on [-pi/2, 0]
I'll be honest here. I have no idea how to do this type of problems. I don't know how to do it. I have these 4 theorems:
1) If f is integrable, then f is bounded
2) If f is defined, bounded and has finite number of disconuities, then f is integrable
3) If f is monotonous, then f is integrable
4) If f is continuous then f is integrable
The thing is, I can solve integrals and no, I'm not lazy and came here just to get a quick solution. This is not the only problem of this type that I've stumbled upon. I just don't know how to solve this type of problems. I don't know how to come up with parameters or functions that can work for given conditions. That's why I'm hoping that I can find something out based on this example so that I can learn to do it.
The best thing would be if there could be some sort of "blueprint" I could guide myself when coming up with the solution for this type of problems but I guess there ain't one. Anyway, if someone can give me a help regarding this problem and similar type problems that would be great. Thanks in advance!