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?
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.