Intuition Behind an Identity

Question

I'm currently studying for a complex analysis prelim. exam in August, so I'm working through some of the exercises in Titchmarsh. One of the exercises has us evaluate the integrals $$\int_0^\infty\frac{1}{1+x^4}\,dx\quad\text{and}\quad\int_0^\infty\frac{x^2}{1+x^4}\,dx.$$After evaluating each of them, I found $$\int_0^\infty\frac{1}{1+x^4}\,dx=\int_0^\infty\frac{x^2}{1+x^4}\,dx=\frac{\pi}{2\sqrt{2}}.$$Pretty sure I had miscalculated, I went to Wolfram Alpha to verify my answers only to find I had done it correctly.

My question is why these two have the same value. Intuitively, I expected $\int\frac{x^2}{1+x^4}\,dx$ to be larger because on the interval $(1,\infty)$, $x^2>1$. The only explanation I can think of is that the $x^2$ makes the integrand much smaller in the interval $[0,1]$ than the original function, but I wouldn't have guessed it to be enough to make the values come out the same. Is there some other intuitive reason why these two integrals are the same?

Answer 1

You may use the change of variables $x\leftrightarrow x^{-1}$ to verify the equality without evaluation.

Answer 2

To mitigate what you found counter-intuitiveness, computing

$$\int_0^1 (1+x^4)^{-1} \, dx \approx 0.867$$ $$\int_1^\infty (1+x^4)^{-1} \, dx \approx 0.244$$

shows that the majority of the area is between $[0,1]$, making it much less surprising that decreasing the value on $[0,1]$ could compensate increasing over $[1,\infty]$.