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  1. $\displaystyle \int_{1}^{\infty} \frac{\ln(x)}{x^2} \, dx.$

  2. $\displaystyle \int_{1}^{\infty} \frac{\ln(x)}{x} \, dx.$

  3. $\displaystyle \int_{0}^{1} \frac{\ln(x)}{\sqrt{x}} \, dx.$

I'm really confused because I've been using comparison test for previous ones such as $1/x^n$ and $x/x^n$ but I don't know what to do when it's $\ln(x)/x^n$. Or any other function such as $\sin x$ in the numerator in that sense. Help on the third one would be awesome as well.

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    $\log(x)$ grows slower than any power of $x$. We have $\lim_{x\to\infty} \frac{\log(x)}{x^r} = 0$ for any $r > 0$ so $\log(x) < x^{r}$ for sufficiently large $x$. This is often useful to solve these kinds of problems. – 2017-02-19

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Hints: 1. $\ln x < x^{1/2}$ for large $x.$

  1. $\ln x >1$ for $x>e.$

  2. $|\ln x| < 1/x^{1/4}$ for small positive $x.$

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    #2 is really giving me trouble, I don't know how to set up the comparison test for it, I can evaluate it and I see it's divergent. Could you clarify? x > e is confusing me. – 2017-02-19
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    Recall $\int_1^\infty (1/x)\,dx=\infty.$ – 2017-02-20