令x=tan t 则：dx=sec^2(t)dt
原式=∫sec^2(t)dt/sec^3(t)=∫cos tdt=sin t+C=sin(arctan x)+C=x/√(x^2+1)+C
验证：[x(x^2+1)^(-1/2)+C]'=(x^2+1)^(-1/2)-1/2*2* x^2 (x^2+1)^(-3/2)
=[x^2+1-x^2](x^2+1)^(-3/2)=1/](x^2+1)^(3/2) 完全正确.sin(arctan x)+C=x/√(x^2+1)+C怎么化得tan(arctan x)=xcot(arctan x)=1/x[csc(arctan x)]^2=1+1/x^2=(x^2+1)/x^2sin(arctan x)+C=x/√(x^2+1)