求不定积分∫(arctanx)/(x^2(x^2+1))dx
问题描述：
求不定积分∫(arctanx)/(x^2(x^2+1))dx
答
求不定积分∫{(arctanx)/[x²(x²+1)]}dx
原式=∫[(arctanx)/x²-(arctanx)/(1+x²)]dx=∫[(arctanx)/x²]dx-∫[(arctanx)/(1+x²)]dx
=-∫(arctanx)d(1/x)-∫(arctanx)d(arctanx)=-{(1/x)arctanx-∫dx/[x(1+x²)]}-(1/2)(arctanx)²
=-(1/x)arctanx+∫[(1/x)-x/(x²+1)]dx-(1/2)(arctanx)²
=-(1/x)arctanx+∫(1/x)dx-(1/2)∫d(x²+1)/(x²+1)-(1/2)(arctanx)²
=-(1/x)arctanx+ln∣x∣-(1/2)ln(x²+1)-(1/2)(arctanx)²+C
答
∫(arctanx)/(x^2(x^2+1))dx=∫(arctanx)/x^2dx-∫(arctanx)/(x^2+1)dx=∫(arctanx)d(1/x)-∫(arctanx)darctanx=arctanx/x-∫1/xdarctanx-1/2(arctanx)^2=arctanx/x-1/2(arctanx)^2-∫1/[x(x^2+1)]dx=arctanx/x-1/2(...