Science:Math Exam Resources/Courses/MATH101/April 2006/Question 03 (d)
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Question 03 (d) 

FullSolution Problems. Justify your answers and show all your work. Simplification of answers is not required. Evaluate the following integral.

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 integrating by parts. 
Hint 2 

Make and . 
Hint 3 

Try integrating by parts again similar to hint 2. 
Hint 4 

Try integrating... no wait the integral on the far right looks like the one on the far left!!! So bring it over and simplify. (This is an example of a cyclic integral). 
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 

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 proceed by integration by parts. Let
and so the integral becomes
We use parts again. Let
Then, we have
This last integral on the right is the same as the integral on the left (up to a constant). Hence, we have
where . This completes the question. 