<s>[INST] Need help calculating integral $\int\frac{dx}{(x^2+4)^2}$ I need some help calculating the integral
$\int\frac{dx}{(x^2+4)^2}$
I tried integration by parts, but I could not arrive to an answer.
 [/INST] As an alternative, note that
$$\int \frac{dx}{(x^2+a^2)^2} = -\frac{1}{2 a} \frac{\partial}{\partial a} \int \frac{dx}{x^2+a^2} = -\frac{1}{2 a} \frac{\partial}{\partial a} \frac{\arctan{(x/a)}}{a}$$
Carry out the differentiation and substitute $a=2$; I get
$$\frac18 \frac{x}{x^2+4} + \frac{1}{16} \arctan{\frac{x}{2}}+C$$
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