calculate-lim-x-0-1-x-n-cos-x-dx-

Question Number 31675 by gunawan last updated on 12/Mar/18

calculate \lim_{x \to \infty} \int_{0}^{1} \frac{x^{n}}{\cos x} d x

Commented by abdo imad last updated on 12/Mar/18

p e r h a p s t h e r e i s a e r r o r i n t h e Q . h o w c a n h a v e 0 ⩽ x ⩽ 1

a n d x \to + \infty ? \dots

Commented by gunawan last updated on 12/Mar/18

I^{'} m sorry Sir, in written like that

Commented by Joel578 last updated on 13/Mar/18

m a y b e \lim_{n \to \infty} \int_{0}^{1} \frac{x^{n}}{\cos x} d x

Commented by abdo imad last updated on 13/Mar/18

l e t u s e t h e c h . x^{n} = t \Leftrightarrow x = t^{\frac{1}{n}} a n d

A_{n} = \int_{0}^{1} \frac{x^{n}}{c o s x} d x = \int_{0}^{1} \frac{t}{c o s (t^{\frac{1}{n}})} \frac{1}{n} t^{\frac{1}{n} - 1} d t

= \frac{1}{n} \int_{0}^{1} \frac{t^{\frac{1}{n}}}{c o s (t^{\frac{1}{n}})} d t = \frac{1}{n} \int_{R} \frac{t^{\frac{1}{n}}}{c o s (t^{\frac{1}{n}})} χ_{[0, 1]} (t) d t t h e

s e q u e n c e o f f u n c t i o n s f_{n} (t) = \frac{t^{\frac{1}{n}}}{c o s (t^{\frac{1}{n}})} χ_{[0, 1]} (t) c . s . t o

χ_{[0, 1]} (t) \Rightarrow {l i m}_{n \to \infty} A_{n} = {l i m}_{n \to \infty} \frac{1}{n} \int_{R} χ_{[0, 1]} (t) d t = 0