Question

How do you find the integral of `f(x)=x^5lnx` using integration by parts?

Answer 1

`I = x^6/6(lnx - 1/6) +C`

Explanation:

`lnx=u => dx/x=du`

`x^5dx=dv => v=x^6/6`

`int udv = uv - int vdu`

`I = int x^5lnxdx = x^6/6lnx - int x^6/6dx/x = x^6/6lnx - 1/6int x^5dx`

`I = x^6/6lnx - 1/6x^6/6 +C`

`I = x^6/6(lnx - 1/6) +C`