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

Evaluate . 
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 

Complete the square in the denominator. 
Hint 2 

Integration by 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. First we complete the square in the expression by considering . This changes the integral to We substitute with and obtain Since the denominator of the last integral has the form we can exploit trigonometric identities when we use the trigonometric substitution . In our case so we substitute . Then and we obtain
where we used the trigonometric identity . The remaining integral is . Now we resubstitute . 