User:Michael Hardy/an integral

Proposition: If

$c=({\text{geometric mean of }}a,b)={\sqrt {ab\,{}}}$

and

$d=({\text{arithmetic mean of }}a,b)={\frac {a+b}{2}}$

then

$\int _{-\infty }^{\infty }{\frac {dx}{\sqrt {(x^{2}+c^{2})(x^{2}+d^{2})}}}=\int _{-\infty }^{\infty }{\frac {dx}{\sqrt {(x^{2}+a^{2})(x^{2}+b^{2})}}}$

Proof: Write the first integral as

$\int _{-\infty }^{\infty }{\frac {du}{\sqrt {(u^{2}+c^{2})(u^{2}+d^{2})}}}$

and then use the substitution

$u={\frac {x-{\frac {ab}{x}}}{2}},\qquad du={\frac {1+{\frac {ab}{x^{2}}}}{2}}\,dx.$

First observe that as u goes from −∞ to +∞, then x goes from 0 to ∞, so the integral becomes

${\begin{aligned}&{}\qquad \int _{0}^{\infty }{\frac {\left({\dfrac {x^{2}+{\frac {ab}{x^{2}}}}{2}}\right)}{\sqrt {\left(\left({\frac {x-{\frac {ab}{x}}}{2}}\right)^{2}+ab\right)\left(\left({\frac {x-{\frac {ab}{x}}}{2}}\right)^{2}+\left({\frac {a+b}{2}}\right)^{2}\right)}}}\,dx\\[10pt]&=\int _{0}^{\infty }{\frac {2(x^{2}+ab)\,dx}{\sqrt {{\Big (}(x^{2}-ab)^{2}+4abx^{2}{\Big )}{\Big (}(x^{2}-ab)^{2}+((a+b)x)^{2}{\Big )}}}}\\[10pt]&=\int _{0}^{\infty }{\frac {2\,dx}{\sqrt {(x^{2}+a^{2})(x^{2}+b^{2})}}}=\int _{-\infty }^{\infty }{\frac {dx}{\sqrt {(x^{2}+a^{2})(x^{2}+b^{2})}}}.\end{aligned}}$