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Question Number 92484 by jagoll last updated on 07/May/20

$$\underset{5}{\stackrel{7}{\int }}{\mathrm{x}}^2\mathrm{f}{}^{‴}\left(\mathrm{x}\right)\mathrm{dx}=?$$$$\mathrm{if}{\mathrm{f}}^{″}\left(7\right)=2,{\mathrm{f}}^{″}\left(5\right)=1$$$${\mathrm{f}}^{\prime }\left(7\right)=-2,{\mathrm{f}}^{\prime }\left(5\right)=-1$$$$\mathrm{f}\left(7\right)=-3,\mathrm{f}\left(5\right)=-4$$$$$$

Answered by john santu last updated on 07/May/20

$$\mathrm{D}.\mathrm{I}\mathrm{method}$$$${\underset{5}{\stackrel{7}{\int }}{\mathrm{x}}^2{\mathrm{f}}^{‴}\left(\mathrm{x}\right)\mathrm{dx}={\mathrm{x}}^2{\mathrm{f}}^{″}\left(\mathrm{x}\right)-2\mathrm{x}{\mathrm{f}}^{\prime }\left(\mathrm{x}\right)+2\mathrm{f}\left(\mathrm{x}\right)]}_5^7$$$$={49\mathrm{f}}^{″}\left(7\right)-{25\mathrm{f}}^{″}\left(5\right)-\{{14\mathrm{f}}^{\prime }\left(7\right)-{10\mathrm{f}}^{\prime }\left(5\right)$$$$+2\mathrm{f}\left(7\right)-2\mathrm{f}\left(5\right)$$$$=98-25-\{-28+10\}+(-6+8)$$$$=73+18+2=93$$

Commented by jagoll last updated on 07/May/20

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