Question

求积分(1/x^2)arctanxdx

Answer 1

积分(1/x^2)arctanxdx：
∫[1-1/(1+x^2)]arctnxdx
=∫arctanxdx-∫arctanxdarctanx
=xarctanx-∫xdx/(1+x^2)-(1/2)(arctanx)^2+C
=xarctanx-(1/2)ln(1+x^2)-(1/2)(arctanx)^2+C

Answer 2

∫(1/x^2)arctanxdx
=-∫arctanxd(1/x)
=-arctanx/x+∫dx/x(1+x^2)
=-arctanx/x+(1/2)∫dx^2/x^2(1+x^2)
=-arctanx/x-(1/2)ln(1+x^2)+ln|x|+C