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求不定积分∫x/(x^2+2x+2)dx

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求不定积分∫x/(x^2+2x+2)dx
求不定积分∫x/(x^2+2x+2)dx

∫x/(x²+2x+2)dx
=1/2∫(2x+2-2)/(x²+2x+2)dx
=1/2∫(2x+2)/(x²+2x+2)dx-∫1/(x²+2x+2)dx
=1/2∫1/(x²+2x+2)d(x²+2x+2)-∫1/[(x+1)²+1]dx
=1/2∫1/udu-∫1/[(x+1)²+1]d(x+1)
=1/2ln|u|-∫1/(u²+1)du
=1/2ln(x²+2x+2)-acrtanu+C
=1/2ln(x²+2x+2)-arctan(x+1)+C