Question

How do you find the integral of `7x^2ln(x) dx`?

Answer 1

I would use Integretion by Parts:

Answer 2

How: Use integration by parts.
(Any integral of the form `ax^nlnx` can be done by parts.)

Details:
`int7x^2ln(x)dx=7intx^2lnxdx`

Let `u=lnx` and `dv=x^2dx`

This makes `du=1/xdx` and `v=intx^2dx=(1/3)x^3` (We'll add the `+C` later.)

`intudv=uv-intvdu`

`7intx^2lnxdx=7[lnx(1/3)x^3-int(1/3)x^3*1/xdx]`

`=7[1/3x^3lnx-1/3intx^3/xdx]=7[1/3x^3lnx-1/3intx^2dx]`

`=7[1/3x^3lnx-1/3(1/3x^3)]+C`

`=7/3x^3lnx-7/9x^3+C`

General
Notice that the solution can be made quite general.
For example: with 5 instead of 3 the problem becomes `intx^5lnxdx` which can be solved by the same reasoning to get ; `1/6x^6lnx-1/36x^6+C`

Note: For even better understanding, check the answers by differentiating.