Question

求不定积分的解题详细步骤！谢谢

1. ∫x^2Inxdx
2. ∫x^2arctanxdx

Answer 1

∫x²lnxdx
解:
令u=lnx,v'=x²,则u'=1/x,v=1/3x³
∫x²lnxdx
=x³/3lnx-1/3∫x²dx
=x³/3lnx-x³/9+C

∫x²arctanxdx
解:
令u=arctanx,v'=x²,则u'=1/x²+1,v=1/3x³
∫x²arctanxdx
=1/3x³arctanx-1/2∫x²/x²+1dx
=1/3x³arctanx-1/2∫x²+1-1/x²+1dx
=1/3x³arctanx-1/2∫(1-1/x²+1)dx
=1/3x³arctanx-1/2(x-arctanx)+C