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

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 = (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`