Science:Math Exam Resources/Courses/MATH103/April 2015/Question 01 (c) (ii)

MATH103 April 2015

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Question 01 (c) (ii)
Determine whether the following integrals converge or diverge. (do not calculate the integrals.) (ii) $\int _{2}^{\infty }{\frac {1}{\ln(x^{10})}}dx$

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 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
Use logarithm properties to simplify the integrand and then apply the Integral Comparison Test.

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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. First, we rewrite the integral as the following $\int _{2}^{\infty }{\frac {1}{\ln(x^{10})}}dx=\int _{2}^{\infty }{\frac {1}{10\ln(x)}}dx$. For any $x>0$, we have $\ln x<x$ which implies that ${\frac {1}{\ln x}}>{\frac {1}{x}}$, therefore, $\int _{2}^{\infty }{\frac {1}{\ln(x)}}dx>\int _{2}^{\infty }{\frac {1}{x}}dx$, since the smaller integral $\int _{2}^{\infty }{\frac {1}{x}}dx$ diverges, hence the bigger one $\int _{2}^{\infty }{\frac {1}{\ln(x)}}dx$ also diverges.