How do I do this integral? (1 Viewer)

Aykay · Sep 30, 2012

Hey guys,

How do I do this integral?

$\int e^xln(x)$

What I do is:

$\int e^xln(x)dx\\ \\ =e^xln(x) - \int e^xx^{-1}dx \\ \\ = e^xln(x) - \left [ e^xx^{-1} + \int e^xx^{-2}dx\right ] \\ \\ = e^xln(x)- \left [ e^xx^{-1} + e^xx^{-2} + 2\int e^xx^{-3}dx \right ] \\ \\ = e^xln(x)- \left [ e^xx^{-1} + e^xx^{-2} + 2e^xx^{-3} + 6\int e^xx^{-4}dx \right ] \\ \\ = e^xln(x)- e^x\left [ \frac{1}{x} +\frac{1}{x^{2}} +\frac{2\times 1}{x^{3}} +\frac{3\times 2\times1}{x^{4}} + ... \right ] \\ \\ = e^x\left [ ln(x) - \left ( \frac{1}{x} +\frac{1!}{x^{2}} +\frac{2!}{x^{3}} +\frac{3!}{x^{4}} + ... \right ) \right]$

No idea what to do from there.
Thanks.

shongaponga · Oct 1, 2012

Basically, the integral $\int {\frac {e^x}{x}} \, dx$ is known to be 'impossible' to integrate in terms of finite combinations of elementary functions. So, this integral is also impossible to integrate in terms of elementary functions.
This is, in fact, the Exponential Integral; Ei, which is a special function on the complex plane. I believe this is out of the realms of the HSC syllabus, hopefully someone can back this up, and perhaps explain it a little better haha. Carrot?

Carrotsticks · Oct 1, 2012

Indeed, the Exponential Integral is required for this question, as it cannot be done via elementary means.

Aykay · Oct 1, 2012

Okay, thanks for that!

How do I do this integral? (1 Viewer)

Aykay

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shongaponga

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Carrotsticks

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Aykay

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