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I need compare these two integrals:

$$ (1) \int_{a}^{b}x^n lnx dx\space \space (2) \int_{a}^{b}x^{n+1} lnx dx $$

for the following values of [a, b]: (A) [1, 2] for both integrals, (B) [0.5, 1] for both integrals and (C) [0.5, 1] for (1) and [0.3, 1] for (2).

What would be the most efficient way to solve this problem? Should I compare $x^nlnx$ and $x^{n+1}lnx$ (and the slope of both functions for a given range) or solve the integrals?

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Hint: When is $x^n \leq x^{n+1}$? When is $\log x \geq 0$? Make a chart and keep track of the signs. –  A Blumenthal Jan 30 '13 at 22:58
    
Integration by parts on the second integral would be a good idea. –  Jp McCarthy Jan 30 '13 at 23:16
    
Another hint: if $g(x) \le f(x)$ on some interval $[a,b]$, then $\int_a^b g(x) \mathrm{d}x \le \int_a^b f(x) \mathrm{d} x.$ –  dirk5959 Jan 30 '13 at 23:29
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1 Answer

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You can give an elegant answer without touching the integrals, using integral properties.

It is know that if $f(x)$ and $g(x)$ are Riemann integrable functions over the closed interval $[a,b]$, and such that $f(x) \geq g(x)$, then $\int_{a}^{b} f(x) dx \geq \int_{a}^{b} g(x) dx$. The same thing goes for $>$ and the opposites.

So you just need to know where your $f(x)$, namely, $x^{n}ln(x)$ is bigger or smaller than $x^{n+1}ln(x)$. Once you know these regions where some condition of order occurs ($f$ is bigger/smaller/equal than $g$), you'll know the relation between both integrals, directly from the properties.

For the C) point of the question, I'd just check if the functions are monotone in the given intervals, if some function is always greater than the other in its corresponding interval, and I would play around with some tricky values for $n$.

Only if I couldn't get any insight from properties or theorems, I would try to compute the integrals.

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