Question

$\displaystyle \int \frac{\left(x+1\right)^{2}}{x\left(x^{2}+1\right)}$ dX is equal to

• A. $log \left|x \left(x^{2}+1\right)\right|+C$
• B. $log \left|x \right|+C$
• C. $log \left|x \right|+2 tan^{-1} x+C$
• D. $log \left(\frac{1}{1+x^{2}}\right)+C$
• E. $2 log\left|x\right|+tan^{-1} x +C$

Answer 1

Answer: (C)

Let $ I =\int \frac{(x+1)^{2}}{x\left(x^{2}+1\right)} d x $
$=\int \frac{x^{2}+1+2 x}{x\left(x^{2}+1\right)} d x $
$=\int \frac{x^{2}+1}{x\left(x^{2}+1\right)} d x+\int \frac{2 x}{x\left(x^{2}+1\right)} d x$
$=\int \frac{1}{x} d x+2 \int \frac{1}{x^{2}+1} d x $
$=\log |x|+2 \tan ^{-1} x+C$