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

Answer 1

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.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

#2#

#I=int_0^(pi/2)sintheta/sqrtcosthetad theta#
Use the substitution #u=costheta#. This implies that #du=-sinthetad theta#.
When we substitute this into the integral, we will have to transform the bounds by plugging the current ones into #costheta#, so the bound of #0# will become #cos(0)=1# and the bound of #pi/2# will become #cos(pi/2)=0#.

Then:

#I=-int_0^(pi/2)(-sintheta)/sqrtcostheta d theta=-int_1^0 1/sqrtudu#

Flipping the integral's bounds with the negative sign and rewriting the exponent:

#I=int_0^1u^(-1/2)du#
Using the rule #intu^ndu=u^(n+1)/(n+1)+C# but using the FTC to evaluate the integral:
#I=[u^(1/2)/(1/2)]_0^1=[2sqrtu]_0^1=2sqrt1-2sqrt0=2#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7