Q. $\int_{}^{}\frac{x}{\left( x^{2} - a^{2} \right)(x^{2} - b^{2})}dx$ is equal to
A) $\frac{1}{a^{2} - b^{2}}\log{\left| \frac{x^{2} - a^{2}}{x^{2} - b^{2}} \right| + c}$
B) $\frac{1}{a^{2} - b^{2}}\log{\left| \frac{x^{2} - b^{2}}{x^{2} - a^{2}} \right| + c}$
C) $\frac{1}{{2(a}^{2} - b^{2})}\log{\left| \frac{x^{2} - a^{2}}{x^{2} - b^{2}} \right| + c}$
D) $\frac{1}{{2(a}^{2} - b^{2})}\log{\left| \frac{x^{2} - b^{2}}{x^{2} - a^{2}} \right| + c}$
Explanation
(c)
Let $I = \int_{}^{}\frac{xdx}{\left( x^{2} - a^{2} \right)(x^{2} - b^{2})}$
Put $x^{2} = t\ \Rightarrow xdx = \frac{1}{2}dt$
$\therefore\ \ \ \ I = \frac{1}{2}\int_{}^{}\frac{dt}{\left( t - a^{2} \right)(t - b^{2})}$
$= \frac{1}{2}\int_{}^{}\left\{ \frac{dt}{\left( a^{2} - b^{2} \right)(t - a^{2})} - \frac{dt}{\left( a^{2} - b^{2} \right)(t - b^{2})} \right\}$
$= \frac{1}{2(a^{2} - b^{2})}\log\left| \frac{t - a^{2}}{t - b^{2}} \right| + c$
$\Rightarrow \ I = \frac{1}{2(a^{2} - b^{2})}\log\left| \frac{x^{2} - a^{2}}{x^{2} - b^{2}} \right| + c\$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.