Why is the integral of this term zero?

Question

I came upon a problem in my physics textbook and had a question as to why term was equal to zero.

The equation and its integration : \begin{align} B &= u(H+I) \\ dB&= u dH + udI \\ \int HdB &= u \int HdH + u \int HdI \end{align} Next, my textbook says that since $\int HdH = 0$, $$\int HdB = u \int HdI.$$

According to my rather basic understanding of Integration, since $\int xdx = x^2/2$, shouldn’t the integration of $HdH$ give $H^2/2$?

Edit: For context,

1. $B = $ Magnetic Induction

2. $H = $ Intensity of Magnetic Field

3. $I = $ Intensity of Magnetisation

4. $ u = $ Magnetic Permeability

   Sorry if the answer is physics based instead of maths.