<s>[INST] Evaluate: $u =\int_0^\infty\frac{dx}{x^4 +7x^2+1}$ Evaluate: 

$u =\displaystyle\int_0^\infty\dfrac{dx}{x^4 +7x^2+1}$

Attempt: 
$$u = \int_0^\infty \dfrac{dx}{\left(x^2+ \left(\dfrac{7 - \sqrt {45}}{2}\right)\right)\left(x^2+ \left(\dfrac{7 + \sqrt {45}}{2}\right)\right)}$$
$$u = \int_0^\infty \dfrac{dx}{(x^2+a^2)(x^2+b^2)}$$
After partial fraction decomposition and simplifying I get: 
$u = \dfrac{\pi}{2(a+b)}$
But answer is $\frac \pi 6$.
Where have I gone wrong? 
 [/INST] I'm not sure you have done anything wrong. You have $a+b$ in your answer, but $a$ is specifically $\sqrt{\frac{7-\sqrt{45}}{2}}=\frac{3-\sqrt{5}}{2}$. And $b=\frac{3+\sqrt{5}}{2}$. So $a+b=3$, making your answer agree with what you are expecting.
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