Find a function $b$ such that the operator $\frac{d}{dx}+b(x)$ is symmetric with the weight $x^2$

Question

Find the value of $b(x) \in \mathbb{C}, x\in \mathbb{R}$, so that $$Â=(Â^{*})^{t}$$ with $$Â=i\frac{d}{dx}+b(x)$$

Here, $(f|g)$ is defined by $$ \int_{-\infty}^{\infty} x^{2}f^{*}(x)g(x)dx $$

I have found $Im{b(x)}=\dfrac{1}{x} $, is it correct?