Question Number 27581 by Mcfaithal last updated on 10/Jan/18
$$\mathrm{If}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\int\left(\mathrm{4}\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\boldsymbol{\mathrm{dx}} \\ $$$$\mathrm{f}\left(\mathrm{3}\right)=\mathrm{22},\:\mathrm{find}\:{f}\left(\mathrm{x}\right) \\ $$
Answered by Joel578 last updated on 10/Jan/18
$${f}\left({x}\right)\:=\:\int\:\mathrm{4}{x}\:−\:{x}^{\mathrm{2}} \:{dx}\:=\:\mathrm{2}{x}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{3}} \:+\:{C} \\ $$$${f}\left(\mathrm{3}\right)\:=\:\mathrm{2}\left(\mathrm{3}^{\mathrm{2}} \right)\:−\:\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{3}^{\mathrm{3}} \right)\:+\:{C}\:=\:\mathrm{22} \\ $$$$\:\rightarrow\:{C}\:=\:\mathrm{22}\:−\:\mathrm{18}\:+\:\mathrm{9}\:=\:\mathrm{13} \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:\mathrm{2}{x}^{\mathrm{2}} \:−\:\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{3}} \:+\:\mathrm{13} \\ $$