Science:Math Exam Resources/Courses/MATH101/April 2013/Question 12 (a)

MATH101 April 2013

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Question 12 (a)
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. Prove that $\displaystyle \int _{2}^{\infty }{\frac {x+\sin x}{1+x^{2}}}\ dx$ diverges.

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
Roughly speaking, the numerator behaves like $x$ whereas the denominator behaves like $x^{2}$, so the integrand should behave like $1/x$, whose integral diverges. To prove this, you don't need to find the exact antiderivative; there are other ways of proving divergence.

Hint 2
Try the comparison test.

Hint 3
What does the integrand look like for large $\displaystyle x$?

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. 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. Since it is difficult to evaluate this integral directly, we try using the comparison test instead to prove divergence. Notice that for large $\displaystyle x$, the integrand behaves like $\displaystyle 1/x$. Since $\int _{2}^{\infty }1/x\ dx$ diverges, we will try to show that our integrand is larger than a constant times $\displaystyle 1/x$. First, since $\displaystyle \sin x\geq -1$, we have $\displaystyle {\frac {x+\sin x}{1+x^{2}}}\geq {\frac {x-1}{1+x^{2}}}\,.$ Now for $\displaystyle x\geq 2$, we have $x-1\geq {\frac {x}{2}}\quad$ and $1+x^{2}\leq 2x^{2}.$ (In fact, the latter is even true for $x\geq 1$.) Hence, over the interval of integration, we have the lower bound ${\frac {x+\sin x}{1+x^{2}}}\geq {\frac {x}{2}}\cdot {\frac {1}{2x^{2}}}={\frac {1}{4x}}\,.$ Finally, since $\displaystyle \int _{2}^{\infty }{\frac {1}{4x}}\,dx={\frac {1}{4}}\int _{2}^{\infty }{\frac {1}{x}}\,dx$ diverges, we conclude that the integral in question diverges as well by the comparison test.

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