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I have the following integral which I want to make sure to solve correctly and transparently: \begin{equation} \int_{\mathbb{R}}\|e^{ax}\|dx \end{equation} If I take cases I obtain: $\int_{\mathbb{R}}\|e^{ax}\|dx=\|\frac{1}{a}e^{ax}\|$ but I don't know how to play with the absolute value inside the integral to obtain a more transparent solution.

i was thinking to move the abolute value like this $\int_{\mathbb{R}}\|e^{ax}\|dx=\int_{\mathbb{R}}e^{\|ax\|}dx$ and go from there but I don't think this is correct.

Any suggestions?

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    $\begingroup$ $e^{ax}$ is negative if.. when is it negative? $\endgroup$
    – Exodd
    Nov 27, 2014 at 17:25

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The function that you are trying to integrate $e^{ax}>0$ for all x in its domain so it is equal to its absolute value for all x in domain.

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  • $\begingroup$ Right, that's a very good point which I missed.I got confused by the fact that the integral can take negative values. But of course I shouldn't care about that. Thanks. $\endgroup$
    – KAT
    Nov 27, 2014 at 18:07

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