Science:Math Exam Resources/Courses/MATH105/April 2011/Question 09 (d)
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Question 09 (d) 

Each of the shortanswer questions below is worth 5 points. Put your answer in the box provided and show your work. No credit will be given for the answer without the correct accompanying work. Determine whether the improper integral is convergent or divergent. If convergent, write the value of the integral in the box. If not, write "divergent" and explain why in the space provided below. 
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 

Check to make sure that the integral makes sense. Where is the integral not defined? 
Hint 2 

An improper integral
with the discontinuity at the point in the interval from to converges if and only if
and
both converge. 
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. We want to determine whether the integral converges. We notice that the integrand is not defined at the point so the integral is improper. Hence, we break up the integral at this discontinuity to get that
To evaluate the first integral, we use a substitution namely and get that . The endpoints change to and
and this last limit evaluates to negative infinity. Hence, the integral diverges. 