I'd like some feedback on my approach to solving this integral

[tomvidm]

1. I gave myself the task of solving the indefinite integral of an exponential function whose base is any real valued constant and whose power is a logarithm of the variable I am integrating over. Now my question is not how to solve it, but rather, whether or not my approach was efficient. I've just started practicing doing the u-substitutions, so my approach was centered around that, but it also involved integration by parts. Below is an image of my paper containing all the information I think is needed.

Was my choice of technique good or could I have solved this integral in some other manner?
(Also, I removed the points 2) and 3) which comes with every thread here. I think the image speaks for itself.)

 

[Infrared]

There is a faster way. Note that $a^{lnx}=(e^{lna})^{lnx}=(e^{lnx})^{lna}=x^{lna}$, which is very easy to integrate. Your answer can be simplified by noting that $e^{lnx}=x$ as long as x>0, and the two answers will be identical.

Edit: This is a minor point, but never forget your constant of integration!