How do you test the improper integral #int sintheta/sqrtcostheta# from #[0,pi/2]# and evaluate if possible?
To test the improper integral ( \int_{0}^{\frac{\pi}{2}} \frac{\sin \theta}{\sqrt{\cos \theta}} ) and evaluate it if possible, you would first check if it converges or diverges using the limit comparison test or another appropriate convergence test for improper integrals. Then, if it converges, you would proceed to evaluate it.
One common approach is to make a trigonometric substitution ( u = \cos \theta ) to simplify the integral. This substitution transforms the integral into a standard form that can be evaluated using techniques such as integration by parts or using trigonometric identities.
After making the substitution and simplifying the integral, you would then evaluate the integral using the limits of integration ( 0 ) and ( \frac{\pi}{2} ). If the integral is divergent, it cannot be evaluated. If it is convergent, you would find the limit as the upper limit of integration approaches ( \frac{\pi}{2} ) to obtain the final result.
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Then:
Flipping the integral's bounds with the negative sign and rewriting the exponent:
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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.
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.
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.
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.
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