Science:Math Exam Resources/Courses/MATH105/April 2013/Question 03 (b)
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Question 03 (b) 

Let and . Calculate . 
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 

Start with the substitution 
Hint 2 

There are two techniques to reduce the order of the derivative under the integral: integration by parts and the fundamental theorem of calculus. You want to use both, in the right order. 
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 begin evaluating this integral by making a change of variable to deal with the square root: And hence, Now using integration by parts we evaluate the integral. Let and so that we have and (which holds by the Fundamental Theorem of Calculus as is an antiderivative of . Thus Now, we once again use the Fundamental Theorem of Calculus and note that is an antiderivative of and this gives Plugging the values of the terms given in the statement of the question gives: 