Question

The value of $\int \frac{d(x^2+1)}{\sqrt{x^2+2}},$ is

• A. $2\sqrt{x^2+2}+c$
• B. $\sqrt{x^2+2}+c$
• C. $x\sqrt{x^2+2}+c$
• D. $4\sqrt{x^2+2}+c$

Answer 1

The correct option is A $2\sqrt{x^2+2}+c$

Consider $I=\int \frac{d(x^2+1)}{\sqrt{x^2+2}}$ we know $d(x^2+1)=2xdx$

$T=\int \frac{2xdx}{\sqrt{x^2+2}}$ put $x^2+2=t\Rightarrow 2xdx=dt$

$I=\int \frac{dt}{\sqrt{t}}=2\sqrt{t}+c\Rightarrow 2\sqrt{x^2+2}+c.$