1. ## Integration Problem (Fractions)

$\displaystyle \int \frac{1}{x^2+12 x + 72}\,dx$

I don't know where to begin! Substitution definitely does not work. Integration by parts & partial fractions do not seem to work either.

I cannot factor the denominator. I get imaginary roots!

How do I proceed with this type of problem? Thanks. PS: I suspect it has something to do with the arctan function!

Thanks!

2. substitution does actually work, the thing is that you need to do some algebra first.

$x^2+12x+72=(x+6)^2+36,$ so put $x+6=6\tan t$ then the integral becomes $\displaystyle\int{\frac{6{{\sec }^{2}}t}{36{{\tan }^{2}}t+36}\,dt}=\frac{1}{6}\int{dt}=\frac{1}{6}t+ k,$ now back substitute.

3. Alternatively, to integrate $\int \frac{1}{(x+ 6)^2+ 36} dx$, let u= (x+ 6)/6 so that du= (1/6) dx, dx= 6du and you have
$\int \frac{1}{36u^3+ 36}(6du)= (1/6)\int \frac{1}{u^2+ 1}du$

which is (1/6)arctan(u)+ C and "back substitution" gives the same answer as Krizalid.

4. i was actually point that direction but when i pushed to button i didn't want to edit.

actually, the latter approach is better than mine since introduces you a known integral that everyone junior should know.