Express the limit limnrarrinfin i1sigman xi ln1 xi2x where

Question

Definite Integrals-

Exercise 1- Express the limit lim_{n→∞ i=1}Σⁿ x_i ln(1 + x_i²)?x, where ?x = 6-2/n and x_i = 2 + i?x, as a definite integral on the interval [2, 6].

Exercise 2- Compute ₀∫²(2x - x³)dx.

Exercise 3- Compute ₀∫⁴e^xdx.

Exercise 4- Prove that _a∫^bx dx = 1/2 (b² - a²). Note that this is f(b) - f(a) where f(x) = ½ x².

Exercise 5- Prove that _a∫^bx² dx = 1/3(b³ - a³). Note that this is f(b) - f(a) where f(x) = 1/3 x³.

Exercise 6- Express the integral ₁∫¹⁰(x - 4 ln x) dx as a limit of Riemann sums as in (1).

Exercise 7- Use the properties of integrals (page 379 to 381) to verify the following inequality without evaluating an integral:

2 ≤ -1∫1√(1 + x²) dx ≤ 2√2.