Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to π/2 of (cot θ dθ)

Accepted Answer

Identify the integral to be tested for convergence: $$\$$int_0$$^{\frac{\pi}{2}} \cot 	heta \, d	heta. $$Recall that $$\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$$, and note that the integrand has potential issues at the endpoints $$	heta = 0$$ and $$	heta = \frac{\pi}{2}$$ because $$\sin 	heta$$ approaches zero there. Split the integral into two improper integrals to analyze the behavior near the problematic points: $$\$$int_0$$^{\frac{\pi}{2}} \cot 	heta \, d	heta = \$$int_0$$^a \cot 	heta \, d	heta + \int_a$$^{\frac{\pi}$${2}} \cot 	heta \, d	heta$$ where $$a is $$some number between 0 and $$\frac{\pi}{2}. $$Use a comparison test by comparing $$\cot 	heta$$ near $$0 to a $$simpler function that behaves like $$\frac{1}{	heta}$$, since $$\sin 	heta \approx 	heta$$ near zero, so $$\cot 	heta \approx \frac{1}{	heta}. $$Similarly, analyze the behavior near $$\frac{\pi}{2}. $$Determine whether the integral converges or diverges by evaluating the limits of the integral near the problematic points or by applying the Direct or Limit Comparison Test with the simpler functions identified.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to π/2 of (cot θ dθ)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.∫ from 0 to π/2 of (cot θ dθ)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to π/2 of (cot θ dθ)