Question

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

Answer 1

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`