Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (c)
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Question 01 (c)  

Determine whether the following integrals converge or diverge. Check appropriate box.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Use the pTest and comparison test for integrals. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution  

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Please rate my easiness! It's quick and helps everyone guide their studies. (i) Since the integrand has an infinite discontinuity at , the integral is defined using a onesided limit, which can be evaluated as follows. Therefore, the integral diverges. (ii) For all positive , we have . Hence the integral converges by comparison to the integral . (iii) For all , we have , which implies that . Therefore , so the integral converges by comparison to the integral . (iv) Since , the integral diverges by comparison to the integral .
