How do you integrate #int x^2 ln 5x dx # using integration by parts?

1 Answer
Bill K.
Feb 17, 2016

Start by letting #u=ln(5x)# and #dv=x^2\ dx# to ultimately get #int \ x^2ln(5x)\ dx=1/3 x^3 ln(5x)-x^3/9+C#.

Explanation:

If you let #u=ln(5x)# and #dv=x^2\ dx#, then #du=1/(5x) * 5\ dx=1/x\ dx# and #v=x^3/3#. Therefore

#\int\ x^2ln(5x)\ dx=uv-\int\ v\ du#

#=1/3 x^3 ln(5x)-\int\ x^2/3\ dx=1/3 x^3 ln(5x)-x^3/9+C#.

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