Integration of f(x)=√(1+x^2) with respect to x^2, is

Let

$I =\int f(x) d\left(x^2\right)=\int \sqrt{1+x^2} d\left(x^2\right)$

$\Rightarrow I =\int \sqrt{1+x^2} d\left(1+x^2\right) ~~~~~\left[∵ d\left(x^2\right)=d\left(1+x^2\right)\right]$

$\Rightarrow I=\frac{2}{3}\left(1+x^2\right)^{3 / 2}+C$