Science:Math Exam Resources/Courses/MATH105/April 2014/Question 01 (f)
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Question 01 (f) 

If and , find . 
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 to find a function such that . 
Hint 2 

Consider integration by parts with and . 
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. We define as . Then and we have:

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. We use integration by parts and set and . Then
Note that the integral on the left and right are the same and thus we can rewrite this as
Dividing by 2 leads to 
Solution 3 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. We use integration by substitution. If then . Thus, . We also can transform the bounds so that and . In terms of u, we can integrate:
