Question

∫xf(x)dx=x^3Inx+C,求不定积分∫f(x)dx

Answer 1

令F(x)=∫f(x)dx
∴∫xf(x)dx=∫xdF(x)=xF(x)-∫F(x)dx=x^3lnx+C
∴∫F(x)dx=xF(x)-x^3lnx+C
两边求导得
F(x)=F(x)+xF'(x)-3x^2lnx-x^3*1/x
=F(x)+xf(x)-3x^2lnx-x^2
f(x)=3xlnx+x

∫f(x)dx

=∫(3xlnx+x)dx
=3∫xlnxdx+∫xdx
=3/2∫lnxdx^2+x^2/2
=3/2x^2lnx-3/2∫x^2*1/xdx+x^2/2
=3/2x^2lnx-3/2∫xdx+x^2/2
=3/2x^2lnx-3/4x^2+x^2/2+C
=3/2x^2lnx-x^2/4+C

Answer 2

对x^3Inx+C求导

得3x^2lnx+x^3*(1/x)=3x^2lnx+x^2
∵∫xf(x)dx=x^3Inx+C

∴xf(x)=3x^2lnx+x^2
f(x)=3xlnx+x

∴∫f(x)dx=∫(3xlnx+x)dx
∫f(x)dx=∫3xlnxdx+∫xdx
分开求∫3xlnxdx和∫xdx
∫3xlnxdx=3∫xlnxdx=3∫lnxd(x^2/2)=3/2x^2lnx-3/2∫x^2d(lnx)=3/2x^2lnx-3/2∫x^2*1/xdx=3/2x^2lnx- 3/2∫xdx=3/2x^2lnx-3/4x^2+C
∫xdx=1/2x^2+C
∴∫f(x)dx=∫3xlnxdx+∫xdx=3/2x^2lnx-1/4x^2+C