Problem 1

We set
$$\begin{array}{rcl}A(k)& =& {\int }_0^{\mathrm{\infty }}\frac{x^2+kx+1}{x^4+1}{\tan }^{-1}\left(\frac{1}{x}\right)dx,\\ B(k)& =& {\int }_0^{\mathrm{\infty }}\frac{x^2+kx+1}{x^4+1}{\tan }^{-1}\left(x\right)dx.\end{array}$$

Solve the followings;

• Prove $A(k)=B(k)$ for any $k\in \mathbf{R}$.

• Solve constants $a,b$ of $\frac{x^2+kx+1}{x^4+1}=\frac{a}{x^2+1+\sqrt{2}x}+\frac{b}{x^2+1-\sqrt{2}x}$ for any $x\in \mathbf{R}$.

• Solve $A(k)$. (hint ${\tan }^{-1}(x)+{\tan }^{-1}(1/x)=\frac{\pi }{2}$ and $x^2+1+\sqrt{2}x=(x+1/\sqrt{2}{)}^2+1/2$).

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• Please comment on your solution, or your solution link.

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