Problem 1

We set
\begin{eqnarray} A(k)\eq\int_0^\infty\f{x^2+kx+1} {x^4+1}\tan^{-1}\left(\f{1}{x}\right)dx,\\ B(k)\eq\int_0^\infty\f{x^2+kx+1}{x^4+1}\tan^{-1}\left({x}\right)dx. \end{eqnarray}

Solve the followings;

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

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

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

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