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

Answer 1

To test the improper integral (\int_0^{\frac{\pi}{2}} \frac{\sin(\theta)}{\cos^2(\theta)} , d\theta), use the substitution (u = \cos(\theta)) to transform the integral into a standard form. Then, evaluate it over the appropriate range. This integral diverges.

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Answer 2

Luke Alvarez

Use limits.

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#.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.