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Question Number 164874 by cortano1 last updated on 22/Jan/22
Answered by mahdipoor last updated on 22/Jan/22
$${get}\:{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}={t}\:\Rightarrow\:{x}^{\mathrm{2}} +\mathrm{1}={t}^{\mathrm{2}} +{x}^{\mathrm{2}} −\mathrm{2}{xt} \\ $$$$\Rightarrow\:\frac{{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{t}}={x} \\ $$$${f}\left({t}\right)=\frac{{x}}{{x}+\mathrm{1}}=\frac{\frac{{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{t}}}{\frac{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{1}}{\mathrm{2}{t}}}=\frac{{t}^{\mathrm{2}} −\mathrm{1}}{{t}^{\mathrm{2}} +\mathrm{2}{t}−\mathrm{1}}\:\: \\ $$