calculate-lim-x-1-x-1-x-2-1-dt-ln-1-t-

Question Number 97981 by abdomathmax last updated on 10/Jun/20

calculate \lim_{x \to 1^{+}} \int_{x - 1}^{x^{2} - 1} \frac{dt}{\ln (1 + t)}

Answered by mathmax by abdo last updated on 11/Jun/20

let F (x) = \int_{x - 1}^{x^{2} - 1} \frac{dt}{\ln (t + 1)} cha 7 gement \ln (t + 1) = u give t + 1 = e^{u}

F (x) = \int_{\ln (x)}^{\ln (x^{2})} \frac{e^{u}}{u} du = \int_{\ln (x)}^{2 \ln (x)} \frac{e^{u}}{u} du \exists c \in] \ln (x), 2 \ln (x) /

F (x) = e^{c} \int_{\ln (x)}^{2 \ln (x)} \frac{du}{u} = e^{c} \ln ∣ \frac{2 lnx}{lnx} ∣ (x \to 1^{+} \Rightarrow c \to 0 +) \Rightarrow \lim_{x \to 1^{+}} F (x) = \ln (2)