Science:Math Exam Resources/Courses/MATH101/April 2018/Question 02 (ii)

MATH101 April 2018

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Question 02 (ii)
Which of the following substitutions is most helpful in evaluating the integral ${\textstyle \displaystyle \int _{2}^{3}{\frac {(x-1)}{\sqrt {4x-x^{2}}}}\,dx}$ ? A: ${\textstyle x=2\sin u}$ B: ${\textstyle x=2+2\sec u}$ C: ${\textstyle x=2+2\tan u}$ D: ${\textstyle x=1+2\sin u}$ E: ${\textstyle x=2+2\sin u}$ F: ${\textstyle x=1+2\tan u}$

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Consider what happens inside the square root with the various substitutions, and how this interacts with the respective differential.

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If we set ${\textstyle x=2+2\sin u}$ , then ${\textstyle dx=2\cos u\;du}$ and ${\sqrt {4x-x^{2}}}={\sqrt {x(4-x)}}={\sqrt {(2+2\sin u)(2-2\sin u)}}={\sqrt {4-4\sin ^{2}u}}=2{\sqrt {1-\sin ^{2}u}}=2\cos u$ so these will cancel leaving just $\int _{0}^{\pi /6}(1+2\sin u)\;du.$ Note that since we do a substitution, we need to change the interval of integration. When $x=2$ , we have $2=2+2\sin u$ so $u=0$ . Similarly, if $x=3$ , we get $u=\pi /6.$ Answer: The correct answer is ${\textstyle {\color {blue}E}}$ .