假如㏒(1/x)是以底為10,真數(shù)為1/x的對(duì)數(shù)
則∫xf(x)dx
=∫x㏒(1/x)dx
=∫xln(1/x)/ln10 dx,換底公式
=(1/ln10)∫xln(x^-1)dx
=(-1/ln10)∫xlnxdx
=(-1/ln10)(1/2)∫lnxd(x²),分部積分
=(-1/2ln10)[x²lnx-∫x²d(lnx)]
=(-1/2ln10)(x²lnx-∫xdx)
=(-1/2ln10)(x²lnx-x²/2)+C
=x²/(4ln10)-x²lnx/(2ln10)+C
亦可以轉(zhuǎn)換為x²/(4ln10)-x²㏒x/2+C