Question

If $\int \frac{xdx}{\sqrt{1+x^2+\sqrt{{(1+x^2)}^3}}}=k\sqrt{1+\sqrt{1+x^2}}+C$ then k equals to

Answer 1

Put, $1+x^2=z^2$
$2zdz=2xdx$
$\int \frac{xdx}{\sqrt{1+x^2+\sqrt{{(1+x^2)}^3}}}$
$=\int \frac{zdz}{\sqrt{z^2+z^3}}$
$=\int \frac{zdz}{z\sqrt{1+z}}=\int \frac{dz}{\sqrt{1+z}}=\int {(1+z)}^{-1/2}\cdot dz$
$=\frac{1}{-1/2+1}\times {(1+z)}^{-1/2+1}+C$
$=\frac{1}{1/2}{(1+z)}^{1/2}+C$
$2\sqrt{1+z}+C$
$=2\sqrt{1+\sqrt{1+x^2}}+C$

Ans: k=2