Question Number 196522 by Erico last updated on 26/Aug/23
$$\mathrm{Prove}\:\mathrm{that}\:\forall\mathrm{n}\in\mathbb{N} \\ $$$$\underset{\:\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} \mathrm{ln}{t}\:\mathrm{dt}\:\leqslant\:\mathrm{ln}\left(\underset{\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} {t}\:\mathrm{dt}\right) \\ $$
Answered by aleks041103 last updated on 26/Aug/23
$$\int{ln}\left({t}\right){dt}\:=\:{tln}\left({t}\right)−{t} \\ $$$$\Rightarrow\int_{{n}} ^{{n}+\mathrm{1}} {ln}\left({t}\right){dt}= \\ $$$$=\left({n}+\mathrm{1}\right){ln}\left({n}+\mathrm{1}\right)−{n}−\mathrm{1}−\left({n}\:{ln}\left({n}\right)−{n}\right)= \\ $$$$=\:{n}\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\:+\:{ln}\left({n}+\mathrm{1}\right)\:−\:\mathrm{1} \\ $$$${ln}\left(\int_{{n}} ^{\:{n}+\mathrm{1}} {tdt}\right)\:=\:{ln}\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)<\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} } \\ $$$$\Rightarrow\int_{{n}} ^{{n}+\mathrm{1}} {ln}\left({t}\right){dt}−{ln}\left(\int_{{n}} ^{\:{n}+\mathrm{1}} {tdt}\right)<−\frac{\mathrm{1}}{\mathrm{2}{n}}+{ln}\left(\frac{{n}+\mathrm{1}}{{n}+\frac{\mathrm{1}}{\mathrm{2}}}\right)= \\ $$$$={ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\right)−\frac{\mathrm{1}}{\mathrm{2}{n}}<\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{2}{n}}<\mathrm{0} \\ $$$$\Rightarrow\int_{{n}} ^{{n}+\mathrm{1}} {ln}\left({t}\right){dt}\leqslant{ln}\left(\int_{{n}} ^{\:{n}+\mathrm{1}} {tdt}\right) \\ $$