Science:Math Exam Resources/Courses/MATH101/April 2007/Question 03 (c)
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Question 03 (c) 

FullSolution Problem. 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 completing the square. 
Hint 2 

Try a trig substitution. 
Hint 3 

Try the trig substitution 
Hint 4 

The integral of should appear. This was one of the trickier integrals we did in the course. What was the trick we used? 
Hint 5 

To evaluate
Try multiplying top and bottom by

Hint 6 

The do the substitution . 
Hint 7 

Don't forget to plug back to the original variables. You will need to draw a triangle and figure out what is in terms of x. 
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 our integral to get Now, let so that . This gives Using the trig identity , we have This last integral has a clever trick. Multiply top and bottom by to get Let so and so the above integral is To get x back in the expression we need to draw our triangle. (Here we used Pythagorean theorem to get the hypotenuse.) From the picture, we see that and so Thus, completing the question. 