let-I-n-0-1-arctan-1-n-1-x-n-find-lim-n-I-n-

Question Number 33845 by prof Abdo imad last updated on 26/Apr/18

l e t I_{n} = \int_{0}^{1} \frac{a r c t a n (1 + n)}{\sqrt{1 + x^{n}}} f i n d {l i m}_{n \to + \infty} I_{n} .

Commented by prof Abdo imad last updated on 27/Apr/18

I_{n} = \int_{R} \frac{a r c t a n (1 + n)}{\sqrt{1 + x^{n}}} χ_{[0, 1]} (x) d x b u t

f_{n} (x) = \frac{a r c t a n (1 + n)}{\sqrt{1 + x^{n}}} χ_{[0, 1]} {(x)}_{n \to + \infty} \to f (x) = \frac{π}{2} o n [0, 1]

s o \int_{R} f_{n} (x) d x \to \int_{0}^{1} f (x) d x = \frac{π}{2} .