<s>[INST] Integrate $\int \frac{6x-15}{x^2 +8x+101}dx$ I've been trying this one for days and I can't seem to get it. Any ideas?
$\int \frac{6x-15}{x^2 +8x+101}dx$
Thanks,
Nick
 [/INST] $\displaystyle\int\dfrac{6x-15}{x^2+8x+101} dx=\int\dfrac{6x-15}{(x+4)^2+85} dx$.
Now let $u=x+4$, so $x=u-4$ and $dx=du$ to get 
$\displaystyle\int\frac{6u-39}{u^2+85} du=3\int\frac{2u}{u^2+85} du-39\int\frac{1}{u^2+85} du$
$=3\ln(u^2+85)-39\left(\frac{1}{\sqrt{85}}\arctan\frac{u}{\sqrt{85}}\right)+C$.
(Now substitute back for u.)
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