<s>[INST] Find the value of $\lim _{a \to \infty} \frac{1}{a} \int_{0}^{\infty} \frac{x^{2}+a x+1}{1+x^{4}} \cdot \tan ^{-1}\left(\frac{1}{x}\right) \,d x $ Find the value of : $$
\lim _{a \rightarrow \infty} \frac{1}{a} \int_{0}^{\infty} \frac{x^{2}+a x+1}{1+x^{4}} \cdot \tan ^{-1}\left(\frac{1}{x}\right) \,d x
$$ I have tried to evaluate this integral by L'Hospitals rule, by separately diffrentiating the numerator and the denominator in the following steps: $$\int_{0}^{\infty} \frac{x^{2}+a x+1}{1+x^{4}} d x\cdot\dfrac{\pi}{2} (\text{by taking x=1/t and adding the integrands)}
$$ I end up converting the above integral in the form $(\text{by Leibnitz's integral rule})$ $$ \dfrac{\pi}{2}
\left(\int_{0}^{\infty} \frac{x}{1+x^{4}} d x\right)
$$ please provide an approach to this problem after this step.
 [/INST] Your solution attempt honestly works just fine as well (and it's also the first thing I thought of when I saw the question!). Indeed from ${x=\frac{1}{t}}$ you get
$${\int_{0}^{\infty}\frac{x^2 + ax + 1}{1+x^4}\arctan\left(\frac{1}{x}\right)dx=\int_{0}^{\infty}\frac{\left(\frac{1}{t}\right)^2 + \frac{a}{t} + 1}{1 + \left(\frac{1}{t}\right)^4}\arctan(t)\frac{1}{t^2}dt}$$
$${=\int_{0}^{\infty}\frac{\left(\frac{1}{t}\right)^4 + \frac{a}{t^3} + \frac{1}{t^2}}{1 + \left(\frac{1}{t}\right)^4}\arctan(t)dt=\int_{0}^{\infty}\frac{1 + at + t^2}{t^4 + 1}\arctan(t)dt}$$
And so
$${2I = \int_{0}^{\infty}\frac{x^2 + ax + 1}{1 + x^4}\left(\arctan(x) + \arctan\left(\frac{1}{x}\right)\right)dx=\frac{\pi}{2}\int_{0}^{\infty}\frac{x^2 + ax + 1}{1 + x^4}dx}$$
So
$${I = \frac{\pi}{4}\int_{0}^{\infty}\frac{x^2 + ax + 1}{x^4 + 1}dx}$$
Now, we don't actually need to use L'hopitals rule. Just notice that
$${\frac{I}{a} = \frac{1}{a}\int_{0}^{\infty}\frac{\frac{\pi}{4}x^2}{x^4 + 1}dx + \frac{\pi}{4}\int_{0}^{\infty}\frac{x}{x^4 + 1}dx + \frac{1}{a}\int_{0}^{\infty}\frac{\frac{\pi}{4}}{x^4 + 1}dx}$$
(We can split the integrals up like this because the integrals all converge separately). As ${a\rightarrow \infty}$, the left-most and right-most terms will obviously go to zero, and the only one that survives is the middle term. Hence we get that the limit is
$${=\frac{\pi}{4}\int_{0}^{\infty}\frac{x}{1+x^4}dx}$$
Now just make the substitution ${u=x^2}$:
$${\Rightarrow \frac{\pi}{8}\int_{0}^{\infty}\frac{1}{1+u^2}du}$$
And finally we are at a standard integral. We know
$${\int_{0}^{\infty}\frac{1}{1+x^2}dx=\frac{\pi}{2}}$$
And so finally we get the limit is
$${\Rightarrow \lim_{a\rightarrow\infty}\frac{I}{a}=\frac{\pi^2}{16}}$$
</s>