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Question Number 169792 by mathocean1 last updated on 08/May/22

f i s c o n t i n u o u s o n R^{+} s u c h t h a t

\int_{0}^{+ \infty} f (t) d t i s c o n v e r g e n t .

D e t e r m i n a t e \underset{x \to + \infty}{l i m} \int_{x}^{x^{2}} f (t) d t .

Answered by aleks041103 last updated on 08/May/22

l e t F (x) = \int_{0}^{x} f (t) d t

t h e n b y p r o b l e m c o n s t r u c t i o n \underset{x \to \infty}{l i m F} (x) = a

\underset{x \to \infty}{l i m} \int_{x}^{x^{2}} f (t) d t = \underset{x \to \infty}{l i m} [F (x^{2}) - F (x)] =

= a - a = 0

\Rightarrow A n s . 0