Science:Math Exam Resources/Courses/MATH101/April 2017/Question 09
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Question 09 

Evaluate . Simplify your answer completely. 
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 

Simplify the integral with the substitution . Then use the integration by parts on the resulting integral. 
Hint 2 

(For the alternative solution) Rewrite as a product so that one of the factors is the derivative of and use integration by parts. 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using the substitution , we have , so that
Then we apply the integration by parts for and to get
Plugging back,
and so
In the last equality, we use the L'hospital's rule to evaluate
Therefore, we get 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. (Alternative solution) Let us rewrite the integrand, so that
Use integration by parts where and so
Note that and Thus, from the first term we have and from the 2nd term we get Answer: 