Question

How do you find the integral of `f(x)=(ln6x)^2` using integration by parts?

Answer 1

The general rule of integration by parts is:

`int u*dv = u*v-int v*du`

Take `u(x) = (ln 6x)^2` and `v(x)=x`

`int (ln6x)^2dx=x(ln 6x)^2 - int x *d ((ln 6x)^2)=x(ln 6x)^2 - int x((2ln 6x)*1/(6x)*6) dx=x(ln 6x)^2-int 2ln6x dx`

Iterating:

`int ln(6x) dx = x ln(6x) -int x d(ln(6x)) = xln(6x) -int x(1/(6x)*6)dx = int dx = xln(6x) - x`

Putting it together:

`int (ln6x)^2dx = x(ln 6x)^2 - 2xln(6x) +2x`