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

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

and

. . . . .[tex]S'(x)\,=\,m_i\,+\, \dfrac{m_{i+1}\, -\, m_i}{h}\, (x\, -\, x_i),\quad x\, \in\, [x_i,\, x_{i+1}][/tex]

so integrate from [tex]x_i[/tex] to [tex]x[/tex]

. . . . .[tex]S(x)\,-\,S(x_i)\, =\, m_i(x-x_i)\, +\, \dfrac{m_{i+1}\, -\, m_i}{h} \, \dfrac{(x\, -\,x_i)(x\, -\, x_i)}{2}[/tex]

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