Hi, I have been working on this 3 hours, and I just can't figure it out. I have this article https://books.google.ee/books?id=5fxu15HQRJ0C&pg=PA60&lpg=PA60&dq=ruutspla in+pedas&source=bl&ots=8h47blCY7y&sig=ce8u4FB5U18A kvFDEyke4ChKnOw&hl=et&sa=X&ved=0ahUKEwiYjuL80dLZAh WFDCwKHf9JCPkQ6AEIJjAA#v=onepage&q=ruutsplain%20pe das&f=false and I need to derive the formula (10) from page 50. Can anyone show me how it's done?

I know that

\[S'(x_i)=m_i, i=0,1,...,n-1\]

and

\[S'(x)=m_i+\frac{m_{i+1}-m_i}{h}(x-x_i), x\in [x_i,x_{i+1}]\]

so integrate from $x_i$ to x

\[S(x)-S(x_i)=m_i(x-x_i)+\frac{m_{i+1}-m_i}{h} \frac{(x-x_i)(x-x_i)}{2}\]