F-x-x-2x-dx-t-4-t-2-1-1-Show-that-F-is-defined-continuous-and-derivable-in-R-

Question Number 128853 by Ar Brandon last updated on 10/Jan/21

F (x) = \int_{x}^{2 x} \frac{dx}{\sqrt{t^{4} + t^{2} + 1}}

1 . Show that F is defined, continuous and derivable in R

Answered by mathmax by abdo last updated on 11/Jan/21

F (x) = \int_{R} \frac{1}{\sqrt{t^{4} + t^{2} + 1}} χ_{[x, 2 x]} (t) dt

the function x \to \frac{χ_{[x, 2 x]} (t)}{\sqrt{t^{4} + t^{2} + 1}} is continue and derivable so

F is derivable and F^{^{'}} (x) = \int_{R} \frac{\partial}{\partial x} {\frac{χ_{[x, 2 x]} (t)}{\sqrt{t^{4} + t^{2} + 1}}} dt

= \frac{2}{\sqrt{{(2 x)}^{4} + {(2 x)}^{2} + 1}} - \frac{1}{\sqrt{x^{4} + x^{2} + 1}} = \frac{2}{\sqrt{{16 x}^{4} + {4 x}^{2} + 1}} - \frac{1}{\sqrt{x^{4} + x^{2} + 1}}