JEE Main 2013 (Online) 23rd AprilQuestion 1
The integral $\int\frac{x\,dx}{2-x^{2}+\sqrt{2-x^{2}}}$ equals :
Indefinite Integration
Solution
Concept. Put $t=\sqrt{2-x^2}$; then $x\,dx$ is $-t\,dt$ and the denominator is $t^2+t$. Formulas used. (1) $t^2=2-x^2\Rightarrow2t\,dt=-2x\,dx$, i.e. $x\,dx=-t\,dt$Step 1 - Substitute. By (1), $$\int\frac{x\,dx}{2-x^2+\sqrt{2-x^2}}=\int\frac{-t\,dt}{t^2+t}=-\int\frac{dt}{1+t}.$$ Step 2 - Integrate. $$-\log|1+t|+c=-\log\Big|1+\sqrt{2-x^2}\Big|+c.$$ Answer: (B)