Question Number 212635 by Nadirhashim last updated on 19/Oct/24
$$\:\:\boldsymbol{{m}}\leqslant\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{1}\:}{\:\sqrt{\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{7}\:}}\:.\boldsymbol{{dx}}\leqslant\boldsymbol{{k}}\:\boldsymbol{{find}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{constant}} \\ $$$$\:\:\boldsymbol{{m}}\:\boldsymbol{{and}}\:\boldsymbol{{k}} \\ $$
Commented by Ghisom last updated on 19/Oct/24
$$\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{strange}\:\mathrm{question} \\ $$$$\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\frac{{dx}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}}}=\sqrt{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right)−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{ln}\:\mathrm{7} \\ $$$$\Rightarrow\:{m}={k}=\sqrt{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{2}}\right)−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{ln}\:\mathrm{7} \\ $$$$\mathrm{if}\:{m},\:{k}\:\in\mathbb{Z}\:\Rightarrow\:{m}=\mathrm{0},\:{k}=\mathrm{1} \\ $$
Answered by mehdee7396 last updated on 19/Oct/24
$${f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}}}\:\:\Rightarrow\:{f}'\left({x}\right)=\frac{−\mathrm{2}{x}}{\:\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}\right)\sqrt{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}}}<\mathrm{0}\Rightarrow{f}\searrow \\ $$$${f}\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{3}}={max}\:\:\:\:\&\:\:\:{f}\left(\mathrm{3}\right)=\frac{\mathrm{1}}{\mathrm{5}}={min} \\ $$$$\Rightarrow\frac{\mathrm{2}}{\mathrm{5}}<\int_{\mathrm{1}} ^{\mathrm{3}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{7}}}{dx}<\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow{m}=\frac{\mathrm{2}}{\mathrm{5}}\:\:\&\:\:{k}=\frac{\mathrm{2}}{\mathrm{3}}\:\: \\ $$$$ \\ $$
Commented by Ghisom last updated on 19/Oct/24
$$\mathrm{this}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{strange}\:\mathrm{answer} \\ $$$$\mathrm{why}\:\mathrm{not} \\ $$$${m}=\frac{\mathrm{13}}{\mathrm{25}}\wedge{k}=\frac{\mathrm{53}}{\mathrm{100}} \\ $$$$\mathrm{or} \\ $$$${m}=\frac{\mathrm{163297}}{\mathrm{312500}}\wedge{k}=\frac{\mathrm{1045101}}{\mathrm{2000000}} \\ $$
Commented by mehdee7396 last updated on 20/Oct/24
Commented by Ghisom last updated on 20/Oct/24
$$\mathrm{but}\:\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{special}\:\mathrm{definition},\:\mathrm{never} \\ $$$$\mathrm{heard}\:\mathrm{of}\:\mathrm{it}\:\mathrm{before}.\:\mathrm{it}\:\mathrm{makes}\:\mathrm{sense}\:\mathrm{for}\:\mathrm{an} \\ $$$$\mathrm{integral}\:\mathrm{we}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{exactly}. \\ $$