<s>[INST] Evaluate $\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$ This is a integration question from a previous calculus exam:

Evaluate $$\int_0^\infty \left( \frac{x^{10}}{1+x^{14}} \right)^{2} \, dx$$

I rewrote it as $$\lim \limits_{b \to \infty} \int_0^b \left(\frac{x^{10}}{1+x^{14}} \right)^{2} \, dx,$$ where do I go on from there?
 [/INST] Hint:
$$\begin{align}\frac{1}{14}\int \frac{x^7 14x^{13}}{(1 + x^{14})^2}dx& \underbrace{=}_{\color{red}{\text{by parts}}}\frac{1}{14}\bigg( -x^7\frac{1}{1 + x^{14}} + 7\color{#05f}{\int \frac{x^6}{1+x^{14}}dx}\bigg) \\&= \frac{1}{14}\bigg(\arctan(x^7) -\frac{x^7}{1 + x^{14}}\bigg) + C\end{align}$$
For $$\color{#05f}{\int \frac{x^6}{1+x^{14}}dx = \frac{1}{7}\arctan (x^7) + C}$$ let $v = x^7$. 
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