Question

求不定积分：∫x(arctanx)^2 dx

Answer 1

=∫(arctanx)²d(x²/2)

=(x²/2)(arctanx)²-∫x²arctanx/(x²+1)dx,设第一项为(*)

=(*)-∫utan²xdu,u=arctanx

=(*)-∫usec²udu+∫udu

=(*)-(utanu-∫tanudu)+u²/2

=(*)-utanu-ln|cosu|+u²/2+C

将u代入即可。