Maths-related interview questions

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Maths-related interview questions

Postby akbar » Tue Jan 29, 2008 7:52 pm

I found this question in a quant test:

integrate f(x)=x^x=exp(x*ln(x)).

Does exp(x*ln(x)) has any primitive? Can't think of one. Tried with MATLAB with no luck.
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Postby mj » Tue Jan 29, 2008 11:13 pm

mathematica can't do it either;

Asking for the derivative of this is more common.
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Postby akbar » Tue Jan 29, 2008 11:17 pm

This question was asked in a quant test, after the derivation one. Do you think it was done to see if the potential candidate is able to recognize when there is no possible solution?
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Postby mj » Tue Jan 29, 2008 11:57 pm

i vaguely recall a brainteaser being to find the definite intergral of this from 0 to 1.

I think was done by expanding e^{x log x} as a power series and then doing it term by term.

If you get this in a written exam, write briefly that a primitive (anti-derivative) does not exist.

In an oral interview, well I 'd say that, and then suggest perhaps getting an approximation term by term.
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Postby INFIDEL » Tue Feb 12, 2008 4:24 pm

The definite integral evaluates to

Sum_{k=1}^infty {(-1)^(k-1)}/k^k
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Re: Maths-related interview questions

Postby salientxu » Mon Jun 21, 2010 10:43 pm

Can be represented using series.
e^{xln(x)}=\sum_{k=0}^{infty}x^n(lnx)^n/n!
Each term can be integrated using linear combinations of x^i(lnx)^j.
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Re:

Postby stochan » Fri Aug 24, 2012 10:31 am

mj wrote:If you get this in a written exam, write briefly that a primitive (anti-derivative) does not exist.


Well, one can always define F(t) to be the integral on [0,t] of the function (which is continuous and therefore Riemann integrable), so I wouldn't write that.

We could maybe say that there is no elementary function whose derivative is x^x (for some definition of elementary) but I do not think they may be interested in such an answer
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