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

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MATH101 April 2010

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Question 03 (b)
Full-Solution Problem. Justify your answer and show all your work. Simplification of the answer is not required. Evaluate the following integral: $\int {\frac {1}{(4+x^{2})^{3/2}}}\,dx$

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
Trig substitution.

Hint 2
Try the substitution $x=2\tan(\theta )$.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Using the change of variable $\displaystyle x=2\tan(\theta )$ and taking a derivative with respect to x on both sides we obtain $\displaystyle 1=2\sec ^{2}(\theta ){\frac {d\theta }{dx}}$ and since $1+\tan ^{2}(\theta )=\sec ^{2}(\theta )$ we have ${\begin{aligned}(4+x^{2})^{3/2}&=(4+4\tan ^{2}(\theta ))^{3/2}\\&=4^{3/2}(1+\tan ^{2}(\theta ))^{3/2}\\&=8(\sec ^{2}(\theta ))^{3/2}\\&=8\sec ^{3}(\theta )\end{aligned}}$ and so our integral becomes ${\begin{aligned}\int {\frac {1}{(4+x^{2})^{3/2}}}\,dx&=\int {\frac {1}{8\sec ^{3}(\theta )}}2\sec ^{2}(\theta )\,d\theta \\&=\int {\frac {1}{4\sec(\theta )}}\,d\theta \\&={\frac {1}{4}}\int \cos(\theta )\,d\theta \\&={\frac {1}{4}}\sin(\theta )+c\end{aligned}}$ Now using that ${\frac {x}{2}}=\tan(\theta )={\frac {\sin(\theta )}{\cos(\theta )}}$ we get that $\cos(\theta )={\frac {2\sin(\theta )}{x}}$ and since $\sin ^{2}(\theta )+\cos ^{2}(\theta )=1$ we have that $\sin ^{2}(\theta )+{\frac {4\sin ^{2}(\theta )}{x^{2}}}=1\quad {\text{so}}\quad \sin ^{2}(\theta )(1+{\frac {4}{x^{2}}})=1$ and so finally we can solve for $\sin(\theta )$ and get $\sin(\theta )={\frac {x}{\sqrt {4+x^{2}}}}$ which we can now plug back into our integral and conclude that $\int {\frac {1}{(4+x^{2})^{3/2}}}\,dx={\frac {x}{4{\sqrt {4+x^{2}}}}}+c$

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