calculate-lim-x-0-x-1-x-2-1-ln-1-t-e-t-dt-

Question Number 39840 by math khazana by abdo last updated on 12/Jul/18

c a l c u l a t e {l i m}_{x \to 0} \int_{x + 1}^{x^{2} + 1} l n (1 + t) e^{- t} d t

Commented by abdo mathsup 649 cc last updated on 12/Jul/18

\exists ? c \in] x + 1, x^{2} + 1 [/

A (x) = \int_{x + 1}^{x^{2} + 1} l n (1 + t) e^{- t} d t = e^{- c} \int_{x + 1}^{x^{2} + 1} l n (1 + t) d t b u t

\int_{x + 1}^{x^{2} + 1} l n (1 + t) d t =_{1 + t = u} \int_{x + 2}^{x^{2} + 2} l n (u) d u

= {[u l n (u) - u]}_{x + 2}^{x^{2} + 2} = (x^{2} + 2) l n (x^{2} + 2) - (x^{2} + 2)

(x + 2) l n (x + 2) + x + 2 \Rightarrow

{l i m}_{x \to 0} \int_{x + 1}^{x^{2} + 1} l n (1 + t) d t = 0 a l s o x \to 0 \Rightarrow c \to 1 \Rightarrow

{l i m}_{x \to 0} A (x) = e^{- 1} . 0 = 0