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Question 04 (b) 

Evaluate the 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 

The substitution u = x^{2} almost works... 
Hint 2 

For an alternative solution, use a 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 1 

For the change of variable u = 1 + x^{2}, we have
So To write the remaining x^{2} in terms of u, we note that since u = 1 + x^{2} we have x^{2} = u − 1. Thus Found a typo? Is this solution unclear? Let us know here, visit the discussion or suggest an alternative solution.

Solution 2 

This integral could be done with a trig substitution as well. We notice inside the square root that we have 1 + x^{2}. This motivates a substitution
so that we can make use of the identity With this substitution we also have Putting this into our integral where we have used the identity from above. This trig integral is of the form with m and n odd. We want to write this in a form that we can easily integrate. If we let then we have hope because we notice the derivative is also there. Making this substitution with we get We now have to substitute back, Therefore we get Found a typo? Is this solution unclear? Let us know here, visit the discussion or suggest an alternative solution.

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