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.

7 posts • Page **1** of **1**

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.

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.

- akbar
**Posts:**28**Joined:**Fri Aug 10, 2007 7:12 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.

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.

- mj
- Site Admin
**Posts:**1380**Joined:**Fri Jul 27, 2007 7:21 am

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.

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.

- salientxu
**Posts:**2**Joined:**Mon Jun 21, 2010 2:58 pm

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

- stochan
**Posts:**4**Joined:**Thu Aug 23, 2012 7:44 am

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