What is the antiderivative of #(ln(x))^3#?

1 Answer
Jan 8, 2016

I found: #xln^3(x)-3xln^2(x)+6xln(x)-6x+c#

Explanation:

We can try to evaluate:
#int(ln(x))^3dx=#
Set #ln(x)=t#
#x=e^t#
#dx=e^tdt#
So we get:
#=intt^3e^tdt=# we can try By Parts:
#=t^3e^t-int3t^2e^tdt=t^3e^t-3intt^2e^tdt=#
Again:
#=t^3e^t-3[t^2e^t-int2te^tdt]=#
#=t^3e^t-3t^2e^t+6intte^tdt=#
Again:
#=t^3e^t-3t^2e^t+6te^t-6inte^tdt=#
#=t^3e^t-3t^2e^t+6te^t-6e^t+c#
But #t=ln(x)#
And remember that #e^(ln(x))=x#. So:
#=xln^3(x)-3xln^2(x)+6xln(x)-6x+c#