Q. $\int_{}^{}{\frac{(x + 1)^{2}}{x(x^{2} + 1)}\ dx}$ is equal to
A) $\log_{e}{x + c}$
B) $\log_{e}{x + 2\tan^{- 1}{x + c}}$
C) $\log_{e}{\frac{1}{x^{2} + 1} + c}$
D) $\log_{e}{\left\{ x\left( x^{2} + 1 \right) \right\} + c}$
Explanation
(b)
$\int_{}^{}{\frac{(x + 1)^{2}}{x(x^{2} + 1)}\ dx} = \int_{}^{}{\frac{x^{2} + 1 + 2x}{x(x^{2} + 1)}dx}$
$= \int_{}^{}{\frac{x^{2} + 1}{x(x^{2} + 1)}dx + 2\int_{}^{}\frac{x}{x(x^{2} + 1)}}dx$
$= \int_{}^{}{\frac{dx}{x} + 2\int_{}^{}\frac{dx}{x^{2} + 1}}$
$= \log_{e}{x + 2\tan^{- 1}{x + c}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.