求不定积分∫(arctanx)/(x^2(x^2+1))dx

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/15 19:24:22

∫(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(arctanx)^2-∫[1/x-x/(x^2+1)]dx
=arctanx/x-1/2(arctanx)^2-lnx+1/2ln(x^2+1)+C

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