Question

Integrate ∫dy/(y(y^2 + 1))

Answer 1

Let us integrate $\int\frac{dy}{y(y^2 + 1)}:$

$\int\frac{dy}{y(y^2 + 1)}=\int(\frac{1}{y}-\frac{y}{y^2+1})dy =\ln|y|-\frac{1}{2}\int(\frac{d(y^2+1)}{y^2+1}) \\=\ln|y|-\frac{1}{2}\ln(y^2+1)+C.$