In Exercises 35–68, use integration, the Direct Comparison Tes...

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

Welcome back everyone. Determine whether the improper integral from 0 to pi divided by 2 of cosi and theta d theta converges or diverges. A converges and b diverges. So for this problem, let's analyze the function that we're integrating, and that's. Coi and Thera. Which is 1 divided by sine theta. So the reason why this is an improper integral is simply because it is singular at theta equals 0, right? Division by 0 is undefined. Let's remember that sin of 0 is 0. And therefore what we can do is rewrite the integral as a limit. Let's call it integral i, and this integral can be written as limit. As A approaches 0 from the right. Of the integral from a to pi divided by 2. Of Kosikan Theradi Thera We get limit as a approaches 0 from the right. Now the integral of cosi and theta d theta is. Negative Alan. Of Cosianhera plus Cotenthera. And we want to evaluate the result from A2 5 divided by 2, right? We can factor out the negative sign, which gives us a negative limit. SA approaches 0 from the right off. And now let's substitute the limits of integration which gives us ln off. Co-se Of pi divided by 2 plus cotangent. Of pi divided by 2. Minus Ln of the absolute value of coequant of a plus. Cotangent of a And now let's simplify. So now, starting with cosequant of I divided by 2. This term is equal to 1, so we can show that it is 1. cotangent of pi divided by 2 is going to be 0. And now as A approaches 0 from the right cosse of. A is going to approach positive infinity. Cotangent of A is going to approach positive infinity as well. So we get negative. In pars LN of 1, that's basically 0. Minus LN of infinity, which is basically infinity as a limit. And negative of negative infinity is positive infinity. Because the result is not a finite number, it is infinity, the integral diverges, and the correct answer is option B. Thank you for watching.

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)

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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)