Science:Math Exam Resources/Courses/MATH101/April 2009/Question 02
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Question 02 

For what values of p does converge? 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Try a substitution and then evaluating the integral directly. 
Hint 2 

Try the substitution (Don't forget to substitute the endpoints!) 
Hint 3 

You might want to isolate the case when 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. First, write this integral as
Let . Then , and . Then
At this point, we see that we need to break this up into two cases. The integral above is different when and when . So suppose . Then we have
This first term will tend to provided that so whenever . This term will tend to infinity when since the logarithmic term stays in the numerator. To take care of the case that we have left out, we have
and this diverges. So our integral converges when and diverges when as required. 