求 $\dps{\int_0^\infty f(x^p+x^{-p}) \frac{\ln x}{1+x^2}\rd x}$ (函数 $f(x)$ 连续) 

解答: $$\beex \bea \mbox{原积分} &=\int_0^1+\int_1^\infty f(x^p+x^{-p}) \frac{\ln x}{1+x^2}\rd x\\ &=\int_\infty^1f(t^{-p}+t^p)\frac{-\ln t}{1+\frac{1}{t^2}}\cdot \sex{-\frac{1}{t^2}}\rd t +\int_1^\infty f(x^p+x^{-p}) \frac{\ln x}{1+x^2}\rd x\quad\sex{x=\frac{1}{t}}\\ &=0. \eea \eeex$$