+26388 Fundamental theorem of calculus
\(\begin{array}{|rcll|} \hline g(s) &=& \displaystyle\int\limits_{8}^{s} f(t)\ dt =\displaystyle \int\limits_{8}^{s} (t-t^4)^5\ dt \\ &=& F(s) - F(8) \\\\ g'(s) &=& \dfrac{d}{ds} \displaystyle\int\limits_{8}^{s} f(t)\ dt \\\\ &=& \dfrac{d\ F(s)}{ds} - \dfrac{d\ F(8)}{ds} \quad &| \quad \dfrac{d\ F(8)}{ds} = 0, \text{ because } F(8) \text{ is constant.} \\\\ &=& \dfrac{d\ F(s)}{ds} \\\\ &=& f(s) \quad & |\quad f(t)=(t-t^4)^5 \\\\ \mathbf{g'(s)}& \mathbf{=} & \mathbf{(s-s^4)^5} \\ \hline \end{array}\)
The Formula is:
\(\begin{array}{|rcll|} \hline \dfrac{d}{ds} \displaystyle\int\limits_{a}^{s} f(t)\ dt = f(s) \\ \hline \end{array}\)
