Question

$\int \frac{xdx}{2-x^2+\sqrt{2-x^2}}$ equals to
(where $C$ is constant of integration)

• A. $\ln |1+\sqrt{2+x^2}|+C$
• B. $-\ln |1+\sqrt{2-x^2}|+C$
• C. $-x\ln |1-\sqrt{2-x^2}|+C$
• D. $x\ln |1-\sqrt{2+x^2}|+C$

Answer 1

The correct option is B $-\ln |1+\sqrt{2-x^2}|+C$

Let $I=\int \frac{xdx}{2-x^2+\sqrt{2-x^2}}$
Put $2-x^2=t^2$
$\Rightarrow xdx=-tdt$
$\therefore I=\int \frac{(-t)dt}{t^2+t}$
$\Rightarrow I=-\int \frac{1}{t+1}dt$
$\Rightarrow I=-\ln |t+1|+C$
$\therefore I=-\ln |\sqrt{2-x^2}+1|+C$