求不定积分∫xarctanxdx

如题所述

∫ x * arctanx dx

用分布积分法 令u=arctanx，v=x ，则∫ udv=uv-∫ vdu
=(1/2)∫ arctanxd(x²)

= ∫ arctanx d(x²/2)
= (x²/2)arctanx - (1/2)∫ x² d(arctanx)
= (x²/2)arctanx - (1/2)∫ x²/(x² + 1) dx
= (x²/2)arctanx - (1/2)∫ (x² + 1 - 1)/(x² + 1) dx
= (x²/2)arctanx - (1/2)∫ dx + (1/2)∫ dx/(x² + 1)
= (x²/2)arctanx - x/2 + (1/2)arctanx + C

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