Question

求不定积分：∫x(arctanx)^2 dx

如题求不定积分∫x(arctanx)^2 dx，请给出详细步骤，不要给图片，因为用手机看不清……

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。

Answer 2

=x^2*(arctanx)^2/2-∫x^2*arctanx/(x^2+1)dx
=x^2*(arctanx)^2/2-∫arctanxdx+∫arctanx/(x^2+1)dx
=x^2*(arctanx)^2/2+(arctanx)^2/2-x*arctanx+∫xdx/(x^2+1)
=(x^2+1)(arctanx)^2/2-x*arctanx+In(x^2+1)/2+C

Answer 3

打出来不容易