Functie integrala

[Beniamin Bogosel]

Fie \( F: (1,\infty)\to \mathbb{R} \) definita prin
\( F(x)=\int_x^{x^2}\frac{{\rm d}t}{\ln t} \).

Demonstrati ca functia este injectiva si gasiti imaginea ei.

IMC 1995

[Vlad Matei]

Hai sa demonstram ca \( \displaystyle \lim_{x\rightarrow 1} F(x)=0 \).Vom folosi \( \displaystyle x-\frac{x^{2}}{2} < \ln{(1+x)} < x \) pentru \( x\in(0;1) \) de unde \( \int_{x}^{x^{2}}\left(\frac{1}{t-1}+\frac{1}{3-t}\right) \hspace{1mm} dt \geq F(x) \geq \int_{x}^{x^{2}} \frac{1}{t-1}\hspace{1mm} dt \) de unde \( \ln{(x+1)}+\ln{(3-x)}-\ln{(3-x^{2})} \geq F(x) \geq \ln{(1+x)} \) si am terminat