Science:Math Exam Resources/Courses/MATH101/April 2010/Question 03 (c)
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Question 03 (c) 

FullSolution Problem. Justify your answer and show all your work. Simplification of the answer is not required. Evaluate the following integral: Note: tan^{1}(x) is also denoted by arctan(x). 
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 

Notice that has an easy derivative. How can we exploit this? 
Hint 2 

Try starting with integration by parts. 
Hint 3 

A clever last trick is to note that 
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. For the integral we start by using integration by parts. Let This gives To evaluate the last integral, we match the denominator by using the fact that . This way we obtain completing the question. 