Science:Math Exam Resources/Courses/MATH101/April 2006/Question 01 (i)

MATH101 April 2006

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Question 01 (i)
Determine whether $\int _{1}^{\infty }{\frac {dx}{1+2x}}$ converges or diverges. If it converges, evaluate it.

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
You can either calculate this directly (Hint 2, Solution 1) or compare it a known p-integral (Hint 3, Solution 2).

Hint 2
If you want to calculate this directly, how do you deal with the upper bound being infinity? Recall that an improper integral like this is the limit of proper (normal) integral.

Hint 3
Comparing the exponents of the polynomial on the top (0) and on the bottom (1) we see that the difference is 1. Hence we suspect that the integral converges or diverges? (Continue at Hint 4)

Hint 4
We expect the integral to diverge, since $\int _{1}^{\infty }{\frac {1}{x}}\,dx$ diverges. Hence, you want to show that ${\frac {1}{1+2x}}\geq C{\frac {1}{x}}$ and then use the integral comparison test.

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 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Evaluating directly, we see that ${\begin{aligned}\int _{1}^{\infty }{\frac {dx}{1+2x}}&=\lim _{b\to \infty }\int _{1}^{b}{\frac {dx}{1+2x}}\\&=\lim _{b\to \infty }\left.{\frac {\ln |1+2x|}{2}}\right|_{1}^{b}\\&=\lim _{b\to \infty }{\frac {\ln |1+2b|}{2}}-{\frac {\ln(3)}{2}}\end{aligned}}$ and this last limit diverges.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Following hints 3 and 4, we want to show that Failed to parse (syntax error): {\displaystyle \frac1{1+2x} \geq C \frac1{x}. } A fraction is larger when the denominator is smaller, hence we can use $1+2x\leq x+2x=3x$ to write ${\frac {1}{1+2x}}\geq {\frac {1}{3x}}.$ Thus, by the integral comparison test, $\int _{1}^{\infty }{\frac {1}{1+2x}}\,dx\geq {\frac {1}{3}}\int _{1}^{\infty }{\frac {1}{x}}\,dx$ However, the latter integral diverges by the integral p-test (p = 1), and thus the former integral also diverges.

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