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I have to integrate the function $$f(x) = \frac{x^2}{\ln(x)} .$$

I have tried to do it by all methods (i.e. by substitution or by parts) but failed.

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    Hint: Let $x=e^u$. – 2017-02-22
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    Not solvable in closed form. [Wolfram Alpha](https://www.wolframalpha.com/input/?i=Integrate%5BX*X%2FLog%5BX%5D,+X%5D) – 2017-02-22
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    It is tantamount to finding $$\int \frac{e^{3y}}y\,dy$$ And that ain't gonna happen anytime soon. – 2017-02-22
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    Maple says the result is $$-{\it Ei} \left( 1,-3\,\ln \left( x \right) \right) $$ – 2017-02-22
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    That's the exponential integral, I still don't think that counts as a closed solution. – 2017-02-22
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    You have to integrate....is this a problem from the book? – 2017-02-22

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Well, you have to integrate: $$\int \frac{x^2}{\ln{x}}~dx$$ We substitute $x=e^u$ and $dx=e^u~du$. This gives: $$\int \frac{e^{3u}}{u}~du \tag{1}$$ This is not solvable in terms of elementary functions, however it can be expressed in terms of the Exponential integral $\operatorname*{Ei}(x)$. Using it's definition: $$\int \frac{e^{3u}}{u}~du=\operatorname*{Ei}(3u)+C$$ Substituting back for $x$ gives the solution.