+26400 integration of 1/(x^2 + 4x + 6) equals what
see: https://www.youtube.com/watch?v=m4SbddauzfU
+118723 integration of 1/(x^2 + 4x + 6) equals what
\(\frac{1}{(x^2 + 4x + 6)}\\ =\frac{1}{(x^2 + 4x + 4)+2}\\ =\frac{1}{(x+2)^2+2}\\ =\frac{1}{2}\frac{1}{\left(\frac{x+2}{\sqrt2}\right)^2+1}\\ \qquad let\\ \qquad y=\frac{x+2}{\sqrt2}\\ \qquad \frac{dy}{dx}=\frac{1}{\sqrt2}\\ \qquad \sqrt2dy=dx\\ =\frac{1}{2}\frac{1}{y^2+1}\\\)
\(\int\frac{1}{x^2 + 4x + 6}\;dx\\ =\int \frac{1}{2}\frac{1}{1+y^2}\;dx\\ =\int \frac{1}{2}\frac{1}{1+y^2}\;\sqrt2dy\\ =\int \frac{\sqrt2}{2}\frac{1}{1+y^2}\;dy\\ =\frac{\sqrt2}{2}tan^{-1}y\\ =\frac{\sqrt2}{2}tan^{-1}\left(\frac{x+2}{\sqrt2}\right)+c\\ \)
