Question

$\int \frac{xdx}{\sqrt{1+x^2+\sqrt{(1+x^2{)}^3}}}$ is

• A. $\frac{1}{2}\ln (1+\sqrt{1+x^2})+c$
• B. $\frac{-2}{3(1+\sqrt{1+x^2}{)}^{\frac{3}{2}}}+c$
• C. $2(\sqrt{1+x^2}+1)+c$
• D. $2\sqrt{1+\sqrt{1+x^2}}+c$

Answer 1

The correct option is D $2\sqrt{1+\sqrt{1+x^2}}+c$

We have,
$\int \frac{xdx}{\sqrt{1+x^2+\sqrt{(1+x^2{)}^3}}}$
$Let1+x^2=t^2\Rightarrow xdx=tdt$
$I=\int \frac{tdt}{\sqrt{t^2+t^3}}=\int \frac{dt}{\sqrt{1+t}}$
$I=2\sqrt{1+t}+c$
$I=2\sqrt{1+\sqrt{1+x^2}}+c$