Dear Uncle Colin,

Why can’t I work out $\int \left( \ln(x) \right)^2 \dx$ using the reverse chain rule?

-- Previously Acceptable, Reasonable Technique Stumbles

Hello, PARTS,

There are two answers to this: the first is, you can’t use the reverse chain rule – which I learned as ‘function-derivative’ when I were a lad – because you don’t have a function and its derivative. I think you’d need an $x$ on the bottom before you went near that.

The second answer is, you can do it by substitution. I don’t like the reverse chain rule as a technique – it’s a special case of substitution, and I’ve always found it to be less error-prone to do the substitution. If you let $u = \ln(x)$, you have $\diff ux = \frac {1}{x}$, and $\d x = \diff xu \d u = e^u \d u$, so your integral becomes $\int u^2 e^u \d u$, which you can do with very little trouble using parts.

I get $u^2 e^u - 2u e^u + 2e^u + C= x \left( (\ln(x))^2 - 2 \ln(x) + 2 \right) + C$.

You can also attack the original integral using parts directly, but I’m not sure how neatly it drops out.

-- Uncle Colin