Question

Express the limit as a definite integral on the given interval. lim n→[infinity] n xi ln(1 + xi2) Δx, [2, 7] i = 1

Answer 1

Answer:

Step-by-step explanation:

$Consider \ the \ following \ series$

$\lim_{n \to \infty} \sum \limits ^n_{i=1} x_i \ In ( 1+ x_i^2) \Delta x , [2,7]$

$Let's \ recall \ the \ explanation \ of$ d$efinite$ $\ integral \ in \ terms \ of \ Riemann \ sum$

$\int^b_a \ f(x) \ dx = \lim_{n \to \infty} \sum \limits ^n_{i=1} \ f(x_i) \Delta x , \ where \ \Delta x = \dfrac{b-a}{n}$

$By \ efinition, \ the \ limit \ is \ then \ expressed \ as \ integral$

$\lim_{n \to \infty} \sum ^a_{i=1} \ x_i \ In (1+x_i^2) \Delta x = \int^7_2 xIn (1+x^2) \ dx$