Question

Evaluate the integral. 6 (x)/(x2 + 4x + 20) dx 2

Answer 1

$\displaystyle\int_{x=2}^{x=6}\frac x{x^2+4x+20}\,\mathrm dx$

$x^2+4x+20=(x+2)^2+16$

Let $x+2=4\tan y$, so that $\mathrm dx=4\sec^2y\,\mathrm dy$. Then the integral becomes

$\displaystyle\int_{y=\pi/4}^{y=\arctan2}\frac{4\tan y-2}{(4\tan y)^2+16}4\sec^2y\,\mathrm dy$
$=\displaystyle\frac12\int_{y=\pi/4}^{y=\arctan 2}(2\tan y-1)\,\mathrm dy$
$=-\ln|\cos y|-\dfrac12y\bigg|_{y=\pi/4}^{y=\arctan2}$
$=\dfrac\pi8+\ln\sqrt{\dfrac52}-\dfrac{\arctan2}2$