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

2 Answers
Mar 8, 2015

I would use Integretion by Parts:
enter image source here

Mar 8, 2015

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.