Question

dy= (2/√(x^2+1)^3)dx

Answer 1

$\displaystyle dy=\frac{2}{\sqrt{x^2+1}^3}dx\\ \\ \\ y=\int \frac{2}{\sqrt{x^2+1}^3}dx\\ \\ \\ \int \frac{2}{\sqrt{x^2+1}^3}dx=\frac{Ax+B}{\sqrt{x^2+1}}+\int \frac{Cx+D}{\sqrt{x^2+1}}dx$

derivando ambos miembros

$\displaystyle \frac{2}{\sqrt{x^2+1}^3}=\frac{Cx^3 + Dx^2 + x(C - B) + A + D}{\sqrt{x^2 + 1}^3}\\ \\ \\ Cx^3 + Dx^2 + x(C - B) + A + D=2\\ \\ \\ \begin{cases} C=0\\ D=0\\ C-B=0\\ A+D=2 \end{cases}\\ \\ A=2\;;\; B=C=D=0$

Entonces

$\displaystyle \int \frac{2}{\sqrt{x^2+1}^3}dx=\frac{2x}{\sqrt{x^2+1}}+C\\ \\ \\ \boxed{y=\frac{2x}{\sqrt{x^2+1}}+C}$