f-x-x-1-x-1-on-1-F-x-

Question Number 128370 by mathocean1 last updated on 06/Jan/21

f (x) = (x - 1) \sqrt{x - 1} o n] 1; + \infty [

F (x) = ?

Answered by mathmax by abdo last updated on 06/Jan/21

if you want derivative f^{^{'}} (x) = \sqrt{x - 1} + (x - 1) \times \frac{1}{2 \sqrt{x - 1}}

= \frac{2 (x - 1) + x - 1}{2 \sqrt{x - 1}} = \frac{2 x - 2 + x - 1}{2 \sqrt{x - 1}} = \frac{3 x - 3}{2 \sqrt{x - 1}}

if you want integral

F (x) = \int f (x) dx = \int (x - 1) \sqrt{x - 1} dx =_{\sqrt{x - 1} = t \to x - 1 = t^{2}} \int t^{2} . t . (2 t) dt

= 2 \int t^{4} dt = \frac{2}{5} t^{5} + C = \frac{2}{5} {(\sqrt{x - 1})}^{5} + C

= \frac{2}{5} {(x - 1)}^{2} \sqrt{x - 1} + C