Question

求不定积分 ∫x^2/(x^2+2x+3)dx

Answer 1

∫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