Science:Math Exam Resources/Courses/MATH101/April 2013/Question 07 (b)

MATH101 April 2013

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Question 07 (b)
Full-Solution Problems. In questions 4–12, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions. Evaluate $\displaystyle \int _{-1}^{1}{\frac {dx}{(x^{2}+1)^{3}}}$. You may use the result of part (a) above, whether or not you completed that problem.

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
Let's try to use part (a). How can we make something that looks like a polynomial turn into something that looks like a trigonometric function?

Hint 2
How do we handle the endpoints?

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We want to be able to use part (a), so we need some way of converting the polynomial into something involving trigonometric functions. In particular, recall that $\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta$ so let's try the substitution $\displaystyle x=\tan \theta$, so $\displaystyle dx=\sec ^{2}\theta \ d\theta$, and the boundaries of integration change according to $\displaystyle x=-1=\tan \theta \Rightarrow \theta =-\pi /4$ and $\displaystyle x=1=\tan \theta \Rightarrow \theta =\pi /4$. Then, using symmetry at the end, as the resulting function is even, we see that $\displaystyle {\begin{aligned}\int _{-1}^{1}{\frac {dx}{(x^{2}+1)^{3}}}&=\int _{-\pi /4}^{\pi /4}{\frac {\sec ^{2}\theta }{(\tan ^{2}\theta +1)^{3}}}\ d\theta \\&=\int _{-\pi /4}^{\pi /4}{\frac {\sec ^{2}\theta }{(\sec ^{2}\theta )^{3}}}\ d\theta \\&=\int _{-\pi /4}^{\pi /4}{\frac {\sec ^{2}\theta }{\sec ^{6}\theta }}\ d\theta \\&=\int _{-\pi /4}^{\pi /4}{\frac {1}{\sec ^{4}\theta }}\ d\theta \\&=\int _{-\pi /4}^{\pi /4}\cos ^{4}\theta \ d\theta \\&=2\int _{0}^{\pi /4}\cos ^{4}\theta \ d\theta \\&=2\left({\frac {8+3\pi }{32}}\right)\\&={\frac {8+3\pi }{16}}\end{aligned}}$

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