A seemingly hard, yet simple integral

When it comes to strange-looking integrals, one must always ask the question: “is it really as strange as it appears to be?”, and then proceed to ask the question: “is there any kind of clever manipulation I can do to the integrand to make this simpler?”. In this particular example, it suffices to note that there is a logarithmic function involved, so one method which comes to mind is the method of subtitution. But substitution of what exactly?

Let us see how the logarithmic function is differentiated. The derivative of ln(x) is, according to standard formulae:

One can take this further, by noting that for any function ln(f(x)), the derivative is

Then, making f(x) = x^n, where n>0, one obtains

This is not surprising given that from logarithmic properties, ln(x^n) = nln(x). If we look at our integrand, it seems like there is something in it that resembles this function. In fact, if we rewrite it as

we can see that the first term in this product is actually the derivative of ln(x^n), divided by n, so more cleverly we could write:

This is fine and all, but we are making things a bit too complicated. All we need to know here is that we can use a substitution. Let’s for instance declare a new variable

And substitute these expressions into our integrand:

This new integral with respect to u is actually quite simple, from standard results we know that

So finally, all we need to do is substitute back u = ln(x^n) into this:

which gives us our final answer.