if-f-2-3-and-f-4-5-find-2-4-f-x-f-x-dx-

Question Number 201464 by hardmath last updated on 06/Dec/23

if f (2) = 3 and f (4) = 5

find \int_{2}^{4} f (x) \cdot f^{^{'}} (x) dx = ?

Answered by mahdipoor last updated on 06/Dec/23

= {[\frac{{(f (x))}^{2}}{2}]}_{2}^{4} = \frac{25 - 9}{2} = 8

Commented by hardmath last updated on 06/Dec/23

thank you, but I {didn}^{'} t understand

Answered by mr W last updated on 07/Dec/23

l e t u = f (x)

d u = f^{'} (x) d x

a t x = 2 : u = f (2) = 3

a t x = 4 : u = f 4) = 5

\int_{2}^{4} f (x) f^{'} (x) d x

= \int_{f (2)}^{f (4)} u d u

= {[\frac{u^{2}}{2}]}_{f (2)}^{f (4)}

= {[\frac{u^{2}}{2}]}_{3}^{5}

= \frac{5^{2} - 3^{2}}{2}

= 8

Commented by Calculusboy last updated on 06/Dec/23

n i c e s o l u t i o n

Commented by hardmath last updated on 08/Dec/23

cool dear professor thank you