Question

How do you test the improper integral `int sintheta/cos^2theta` from `[0,pi/2]` and evaluate if possible?

Answer 1

Use limits.

Explanation:

The integral is improper where `costheta = 0`, which occurs at `theta = pi/2`.
To evaluate,
`int_0^(pi/2) sintheta/cos^2theta d(theta)`
first observe that `sintheta/cos^2theta = sec(theta)tan(theta)`
Use a limit on the improper end:
`int_0^(pi/2) sec(theta)tan(theta) d(theta) = lim(int_0^bsec(theta)tan(theta) d(theta) )`, where the limit is as `b -> (pi/2)^-`
Evaluate:
`lim(int_0^b sec(theta)tan(theta) d(theta) ) =`
`= lim(secb - sec0)`
`= lim(secb - 1)`
This limit diverges as `b -> pi/2`.