令x=tant
则dx=sec^2tdt
于是
∫dx/[x(x^2+1)]
=∫sec^2t/[tantsec^2t]dt
=∫dt/tant
=∫(cost/sint)dt
=∫(1/sint)dsint
=ln|sint|+C
三角替换sint=x/√(1+x^2)
所以∫dx/[x(x^2+1)]=ln|x/√(1+x^2)|+C