Q. If a function $f(x)$ satisfies $f^{'}(x) = g(x)$ Then, the value of $\int_{a}^{b}{f(x)g(x)dx}$ is
A) $\frac{1}{2}\left\lbrack \left( f(b) \right)^{2} - \left( F(a) \right)^{2} \right\rbrack$
B) $\frac{1}{2}\left\lbrack \left( f(b) \right)^{2} + \left( f(a) \right)^{2} \right\rbrack$
C) $\frac{1}{2}\left\lbrack f(b) - f(a) \right\rbrack^{2}$
D) None of these
Explanation
(a)
$\because f^{'}(x) = g(x)$
$\Rightarrow \int_{a}^{b}{f(x)g(x)dx = \int_{a}^{b}{f(x)f^{'}(x)dx}}$
$= \left\lbrack \frac{\left( f(x) \right)^{2}}{2} \right\rbrack_{a}^{b}$
$= \frac{1}{2}\lbrack\left( f(b) \right)^{2} - \left( f(a) \right)^{2}\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.