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Find the indefinite integral

$$\int\frac{(x+1)e^x}{x(1+xe^x)}dx$$

I feel like this function does not have an anti-derivative in the form of elementary functions.

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  • $\begingroup$ So no one else has to try typing it in... Wolfram Alpha doesn't find anything. $\endgroup$ – apnorton Nov 26 '13 at 17:03
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    $\begingroup$ @Katoyu, I think, the multiplicand $x$ in the denominator is a typo as $$\frac{d(1+xe^x)}{dx}=(x+1)e^x$$ $\endgroup$ – lab bhattacharjee Nov 26 '13 at 17:05
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You're entirely correct that the posted problem has no elementary antiderivative; you can easily confirm this using Wolfram Alpha.

However, if perchance your problem has a typo in it, and you meant to post:

$$\int\frac{(x+1)e^x}{(1+xe^x)}dx$$

Then note that with

  • $\;u = 1 + xe^x,\;$

  • $\;du = xe^x + e^x = (x+1)e^x$,

    thus giving us an integral of the form $$\int \dfrac {du}{u} = \ln |u| + C = \ln|1 + xe^x| + C$$

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You're right: Maple confirms that it does not have an elementary antiderivative.

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As written, you're correct. However, I suspect that it is supposed to be $$\int\frac{(x+1)e^x}{1+xe^x}\,dx,$$ which does have a nice antiderivative family, as the numerator is the derivative of the denominator.

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