Question

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

Answer 1

`x((ln(x))^3-3(ln(x))^2+6ln(x)-6)+C`

Explanation:

`I=int(ln(x))^3dx`

Let `t=ln(x)`. This implies that `x=e^t` so `dx=e^tdt`. Then:

`I=intt^3(e^tdt)`

On this, we can perform integration by parts a number of times.

`{(u=t^3,=>,du=3t^2dt),(dv=e^tdt,=>,v=e^t):}`

`I=t^3e^t-int3t^2(e^tdt)`

`{(u=3t^2,=>,du=6tdt),(dv=e^tdt,=>,v=e^t):}`

`I=t^3e^t-(3t^2e^t-int6t(e^tdt))`

`I=t^3e^t-3t^2e^t+int6t(e^tdt)`

`{(u=6,=>,du=6dt),(dv=e^tdt,=>,v=e^t):}`

`I=t^3e^t-3t^2e^t+6te^t-int6e^tdt`

`I=t^3e^t-3t^2e^t+6te^t-6e^t+C`

`I=e^t(t^3-3t^2+6t-6)+C`

Using `t=ln(x)` and `e^t=x`:

`I=x((ln(x))^3-3(ln(x))^2+6ln(x)-6)+C`