Question

计算定积分∫e(在上)1(在下)x平方lnxdx(在中间)

Answer 1

∫x²lnxdx
=1/3∫lnxdx³
=1/3*lnx*x³-1/3∫x³dlnx
=1/3*lnx*x³-1/3∫x³*1/xdx
=1/3*lnx*x³-1/3∫x²dx
=1/3*lnx*x³-1/9*x³+C

所以定积分=e³/3-e³/9-(0-1/9)
=(2e³+1)/9

Answer 2

∫[1,e]x^2lnxdx=∫[1,e]lnxd(x^3/3)=lnx*x^3/3|上e下1-∫[1,e]x^2/3 dx
=e^3/3-x^3/9|上e下1=e^3/3-e^3/9+1/9=2e^3/9+1/9