integration of 1/(x^2 + 4x + 6) equals what

integration of 1/(x^2 + 4x + 6) equals what

see: https://www.youtube.com/watch?v=m4SbddauzfU

As below:

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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\\ $