Science:Math Exam Resources/Courses/MATH101/April 2005/Question 03 (c)
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Question 03 (c) 

FullSolution Problem. Justify your answer and show all your work. Simplification of your answer 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 to modify the integrand in order to perform a trigonometric substitution. 
Hint 2 

Complete the square in the denominator and make an appropriate trigonometric substitution. 
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 integrate: The first step is to complete the square in the denominator: Now we substitute: So the integral simplifies to We now perform a trigonometric substitution (this step is only valid for y smaller than 1, which we know is true because 1y^2 must be positive): So the above integral becomes: But for any u, we have So the above quantity is equal to So we have
