Background: A friend once brought up the integral $\displaystyle\int_0^\infty\frac{1}{1+x^4}\,dx$ and asked me if I can solve it. Now, I converted it into a beta function and solved it using Euler's reflection property (Evaluated to $\frac{\pi}{2\sqrt2}$). But my friend and I are in high school and he didn't know the use of beta function  or most of the techniques that I had used there. He then challenged me to solve it using elementary high school metheds. My Efforts: I thought that since I already knew the value it was going to evaluate  to, I could brute-force my way to find this. I thought to use the sandwich theorem and take the bounds such that they evaluate to $\frac{\pi}{2\sqrt2}$. For the upper bound, I thought that since I want $\pi$ in it I could use the derivative of $\arctan(x)$ since it is also similar to function we are trying to integrate. $$\int_0^\infty \frac{1}{\sqrt2} \frac{1}{(x^2 + 1)}\,dx \geq\int_0^\infty \frac{1}{x^4+1}  $$ Now, the problem here is that I can't exactly prove that this inequality is true. I can definitely say that after a point $x\approx 1.3$ , $\displaystyle \frac{1}{\sqrt2}\frac{1}{1+x^2} &gt; \frac{1}{1+x^4}$. But then from this I can't really comment on the inequality of the above integrals. Also I can't find a suitable lower bound for this. My Question:  Can we prove this inequality mentioned above, also what would be a suitable lower bound for this? Otherwise are there any better bounds I can use here (all knowing what the integral is going to converge to already) ? Or should I accept defeat as there is &quot;no way&quot; to solve this using elementary high school methods?  Any input appreciated..
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help.
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