[tex]\displaystyle\int_{x=2}^{x=6}\frac x{x^2+4x+20}\,\mathrm dx[/tex]
[tex]x^2+4x+20=(x+2)^2+16[/tex]
Let [tex]x+2=4\tan y[/tex], so that [tex]\mathrm dx=4\sec^2y\,\mathrm dy[/tex]. Then the integral becomes
[tex]\displaystyle\int_{y=\pi/4}^{y=\arctan2}\frac{4\tan y-2}{(4\tan y)^2+16}4\sec^2y\,\mathrm dy[/tex]
[tex]=\displaystyle\frac12\int_{y=\pi/4}^{y=\arctan 2}(2\tan y-1)\,\mathrm dy[/tex]
[tex]=-\ln|\cos y|-\dfrac12y\bigg|_{y=\pi/4}^{y=\arctan2}[/tex]
[tex]=\dfrac\pi8+\ln\sqrt{\dfrac52}-\dfrac{\arctan2}2[/tex]
