lnx除以x的不定积分怎么求(不定积分xa)
不定积分∫x^a(lnx)^2dx的计算
主要内容:
通过多次分部积分法介绍不定积分∫x^a(lnx)^2dx的计算步骤。
通式推导:
∫x^a(lnx)^2dx
=1/(a 1)∫(lnx)^2dx^(a 1),以下第一次使用分部积分法,
=1/(a 1) (lnx)^2*x^(a 1) -1/(a 1)∫x^(a 1) d(lnx)^2
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)∫x^(a 1) *lnx*(1/x)dx
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)∫x^a*lnxdx
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2∫lnxdx^a11,以下第二次使用分部积分法,
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^2∫x^(a 1) dlnx
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^2∫x^(a 1) *1/xdx
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^2∫x^adx
=1/(a 1) (lnx)^2*x^(a 1) -2/(a 1)^2lnx*x^(a 1) 2/(a 1)^3x^(a 1) c
=x^(a 1) [(lnx)^2/(a 1) -2/(a 1)^2lnx 2/(a 1)^3] c
举例计算,例如当a=4时,计算过程如下:
∫x^4 (lnx)^2dx
=(1/5)∫(lnx)^2dx^a11,以下第一次使用分部积分法,
=(1/5) (lnx)^2*x^5-(1/5)∫x^5d(lnx)^2
=(1/5) (lnx)^2*x^5-(2/5)∫x^5*lnx*(1/x)dx
=(1/5) (lnx)^2*x^5-(2/5)∫x^4*lnxdx
=(1/5) (lnx)^2*x^5-(2/25)∫lnxdx^5,以下第二次使用分部积分法,
=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/25)∫x^5dlnx
=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/25)∫x^5*1/xdx
=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/25)∫x^adx
=(1/5) (lnx)^2*x^5-(2/25)lnx*x^5 (2/125)x^5 c
=x^5 [(1/5) (lnx)^2-(2/25)lnx (2/125)] c
=(1/125)x^5 [25 (lnx)^2-10lnx 2] c.
,免责声明:本文仅代表文章作者的个人观点,与本站无关。其原创性、真实性以及文中陈述文字和内容未经本站证实,对本文以及其中全部或者部分内容文字的真实性、完整性和原创性本站不作任何保证或承诺,请读者仅作参考,并自行核实相关内容。文章投诉邮箱:anhduc.ph@yahoo.com