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