Science:Math Exam Resources/Courses/MATH103/April 2013/Question 02 (a)
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Question 02 (a) 

Integration: Short Answer Problems  Evaluate the following integrals: state if a definite integral does not exist; use limits for improper integrals. (full marks for correct answer; work must be shown for partial marks)

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. 
Hint 2 

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

(Alternate Solution) Try 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. Let so that , and . Then, we have
completing the problem. 
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. Proceed by integration by parts. Let and . Then and so applying integration by parts gives
As now appears on both sides, we bring the value over to get or that completing the problem. 