How do you integrate #int(lnx)^2/x# by integration by parts method?
3 Answers
Explanation:
Integration by parts is not the best way to solve this integral.
As
and have:
I got:
Explanation:
We have:
so that basically we have that:
now a trick...
take the last integral to the left of the equal sign as in a normal equation:
add the two integrals and rearrange:
Explanation:
#I=int(lnx)^2/xdx#
Integration by parts is not necessary. The quickest way to do this is with the substitution
#I=int(lnx)^2(1/xdx)=intu^2du=1/3u^3=1/3(lnx)^3+C#
We can do integration by parts, however, letting:
#{(u=(lnx)^2,=>,du=(2lnx)/xdx),(dv=1/xdx,=>,v=lnx):}#
Then:
#I=uv-intvdu#
#I=(lnx)^3-2int(lnx)^2/xdx#
This is the original integral:
#I=(lnx)^3-2I#
#3I=(lnx)^3#
#I=1/3(lnx)^3+C#