Science:Math Exam Resources/Courses/MATH101/April 2013/Question 12 (a)
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Question 12 (a) 

FullSolution Problems. In questions 4–12, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions. Prove that diverges. 
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 

Roughly speaking, the numerator behaves like whereas the denominator behaves like , so the integrand should behave like , whose integral diverges. To prove this, you don't need to find the exact antiderivative; there are other ways of proving divergence. 
Hint 2 

Try the comparison test. 
Hint 3 

What does the integrand look like for large ? 
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. Since it is difficult to evaluate this integral directly, we try using the comparison test instead to prove divergence. Notice that for large , the integrand behaves like . Since diverges, we will try to show that our integrand is larger than a constant times .
Now for , we have and (In fact, the latter is even true for .) Hence, over the interval of integration, we have the lower bound Finally, since diverges, we conclude that the integral in question diverges as well by the comparison test. 