∫x²/(x²+2x+3)dx
=∫(x²+2x+3-2x-3)/(x²+2x+3)dx
=∫dx-∫(2x+3)/(x²+2x+3)dx
=x-∫(2x+2+1)/(x²+2x+3)dx
=x-∫(2x+2)/(x²+2x+3)dx-1/(x²+2x+3)dx
=x-∫d(x²+2x+3)/(x²+2x+3)-1/[(x+1)²+2]dx
=x-ln(x²+2x+3)-d(x+1)/[(x+1)²+2]
=x-ln(x²+2x+3)-1/√2arctan(x+1)/√2+C
∫x²/(x²+2x+3)dx
=∫(x²+2x+3-2x-3)/(x²+2x+3)dx
=∫dx-∫(2x+3)/(x²+2x+3)dx
=x-∫(2x+2+1)/(x²+2x+3)dx
=x-∫(2x+2)/(x²+2x+3)dx-1/(x²+2x+3)dx
=x-∫d(x²+2x+3)/(x²+2x+3)-1/[(x+1)²+2]dx
=x-ln(x²+2x+3)-d(x+1)/[(x+1)²+2]
=x-ln(x²+2x+3)-1/√2arctan(x+1)/√2+C