Integration Questions

Question Number 183450 by cortano1 last updated on 26/Dec/22

$$\:\underset{\mathrm{0}} {\overset{{x}^{\mathrm{2}} } {\int}}\:{f}\left(\mathrm{2}{t}^{\mathrm{3}} −{t}^{\mathrm{2}} −\mathrm{6}\right){dt}\:=\:\mathrm{ln}\:{x}\: \\$$ $$\:{then}\:{f}\left(\mathrm{106}\right)=? \\$$

Answered by mahdipoor last updated on 26/Dec/22

$$\frac{{d}}{{dx}}\:{L}=\frac{{d}}{{dx}}\:{R}\:\Rightarrow\:{f}\left(\mathrm{2}{x}^{\mathrm{6}} −{x}^{\mathrm{2}} −\mathrm{6}\right)×\mathrm{2}{x}=\frac{\mathrm{1}}{{x}} \\$$ $${x}=\mathrm{2}\:\Rightarrow\:{f}\left(\mathrm{106}\right)=\frac{\mathrm{1}}{\mathrm{8}} \\$$ $$−−−−−−− \\$$ $$\frac{{d}}{{dx}}\:\int_{\mathrm{0}} ^{\:{g}\left({x}\right)} {f}\left({h}\left({t}\right)\right){dt}=\frac{\int_{{g}\left({x}\right)} ^{\:{g}\left({x}+\Delta{x}\right)} {f}\left({g}\left({t}\right)\right){dt}}{\Delta{x}}\:\Delta{x}\rightarrow\mathrm{0} \\$$ $$\Rightarrow\left({get}\:\int{f}\left({h}\left({t}\right)\right){dt}={k}\left({t}\right)\:\right)\:\Rightarrow \\$$ $$=\frac{{k}\left({g}\left({x}+\Delta{x}\right)\right)−{k}\left({g}\left({x}\right)\right)}{\Delta{x}}={k}^{'} \left({g}\left({x}\right)\right){g}^{'} \left({x}\right) \\$$ $$={f}\left({h}\left({g}\left({x}\right)\right)\right){g}^{'} \left({x}\right) \\$$ $$\Rightarrow{h}\left({t}\right)=\mathrm{2}{t}^{\mathrm{3}} −{t}^{\mathrm{2}} −\mathrm{6}\:\:\:\:\:\:{g}\left({x}\right)={x}^{\mathrm{2}} \:\:\:\Rightarrow \\$$ $${f}\left(\mathrm{2}{x}^{\mathrm{6}} −{x}^{\mathrm{4}} −\mathrm{6}\right)×\mathrm{2}{x} \\$$ $$\\$$

Answered by mr W last updated on 26/Dec/22

$${say}\:{f}\left(\mathrm{2}{t}^{\mathrm{3}} −{t}^{\mathrm{2}} −\mathrm{6}\right)={g}\left({t}\right) \\$$ $$\int_{\mathrm{0}} ^{{x}^{\mathrm{2}} } {g}\left({t}\right){dt}=\mathrm{ln}\:{x} \\$$ $$\frac{{d}}{{dx}}\left[\int_{\mathrm{0}} ^{{x}^{\mathrm{2}} } {g}\left({t}\right){dt}\right]=\frac{{d}}{{dx}}\left(\mathrm{ln}\:{x}\right) \\$$ $${g}\left({x}^{\mathrm{2}} \right)\left(\mathrm{2}{x}\right)=\frac{\mathrm{1}}{{x}} \\$$ $$\Rightarrow{g}\left({x}^{\mathrm{2}} \right)=\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} } \\$$ $$\Rightarrow{g}\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}{t}} \\$$ $$\Rightarrow{f}\left(\mathrm{2}{t}^{\mathrm{3}} −{t}^{\mathrm{2}} −\mathrm{6}\right)=\frac{\mathrm{1}}{\mathrm{2}{t}} \\$$ $${set}\:\mathrm{2}{t}^{\mathrm{3}} −{t}^{\mathrm{2}} −\mathrm{6}=\mathrm{106} \\$$ $$\mathrm{2}{t}^{\mathrm{3}} −{t}^{\mathrm{2}} −\mathrm{112}=\mathrm{0} \\$$ $$\left({t}−\mathrm{4}\right)\left(\mathrm{2}{t}^{\mathrm{2}} +\mathrm{7}{t}+\mathrm{28}\right)=\mathrm{0} \\$$ $$\Rightarrow{t}=\mathrm{4} \\$$ $$\Rightarrow{f}\left(\mathrm{106}\right)=\frac{\mathrm{1}}{\mathrm{2}×\mathrm{4}}=\frac{\mathrm{1}}{\mathrm{8}}\:\checkmark \\$$