Question

大一高数积分题第七题

• 你的回答被采纳后将获得：
• 系统奖励15（财富值+成长值）+难题奖励20（财富值+成长值）

Answer 1

被积函数=A/(x+1) + B/(x+2) + C/(x+3)。
用待定系数法把A，B，C确定出来然后积分。

Answer 2

1/(x+1)(x+2)(x+3)=1/(x+1)[1/(x+2)-1/(x+3)]
=1/[(x+1)(x+2)]-1/[(x+1)(x+3)]
=1/(x+1)-1/(x+2)-1/2[1/(x+1)-1/(x+3)]
=1/[2(x+1)]-1/(x+2)+1/[2(x+3)]

∫x/(x+a)dx=∫[1-a/(x+a)]dx=x-aln|x+a|+C

∫x/(x+1)(x+2)(x+3)dx
=∫x/[2(x+1)]-x/(x+2)-x/[2(x+3)]dx
=1/2∫x/(x+1)dx-∫x/(x+2)dx+1/2∫x/(x+3)dx
=1/2(x-ln|x+1|)-(x-2ln|x+2|)+1/2(x-3ln|x+3|)+C
=-1/2ln|x+1|+2ln|x+2|-3/2ln|x+3|+C