A cute log gamma integral

Hopefully you enjoy the problem and before looking at the solution, give it a go!

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Looking at the integral, we may be clueless with what to do at first since we are used to seeing the integral version of gamma function. However, in situations like these, we often want to exploit properties of the gamma function instead. So let us recall Euler’s reflection formula:

This is clearly quite favourable to use looking at our given limits. So, let us try the substitution u = 1-x to give us:

These have the same limits so we can add them together. If our integral is I, then we can write this as:

Now, we can see where we can start using Euler’s reflection formula! This gives us:

Which is the same as the following:

Performing the substitution u=1-x, we can see that its symmetric hence we can write this as the following:

And hence

Which is precisely an integral we have seen before here! Now, that means that this is nothing but

Which gives the beautiful result: