Q. If $\int_{}^{}{f(x)dx = g(x)}$, then $f(x)g(x)$ is equal to
A) $\frac{1}{2}f^{2}(x)$
B) $\frac{1}{2}g^{2}(x)$
C) $\frac{1}{2}\left\lbrack g^{'}(x) \right\rbrack^{2}$
D) $f^{'}(x)g(x)$
Explanation
(b)
Given, $\int_{}^{}{f(x)dx = g(x)}$
$\therefore\int_{}^{}{f(x)g(x)}dx = g(x)\int_{}^{}{f(x)}dx - \int_{}^{}{\lbrack g^{'}(x)}\int_{}^{}{f(x)}\ dx\rbrack\ dx$
$= g(x)g(x) - \int_{}^{}{g^{'}(x)}g(x)dx - \left\lbrack g(x) \right\rbrack^{2} - \int_{}^{}{f(x)}g(x)dx$
$\Rightarrow \int_{}^{}{f(x)}g(x)dx = \frac{1}{2}\left\lbrack g(x) \right\rbrack^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.