FIND int_. (x^2)ln 5x dx using IBP (Integration by parts) ?

1 Answer
Mar 8, 2018

I=1/9x^3(3ln(5x)-1)+C

Explanation:

We want to integrate

I=intx^2ln(5x)dx

Use integration by parts

intudv=uv-intvdu

Let u=ln(5x) and dv=x^2dx

Then du=1/xdx and v=1/3x^3

Thus

I=ln(5x)1/3x^3-int1/3x^3*1/xdx

color(white)(I)=1/3ln(5x)x^3-1/3intx^2dx

color(white)(I)=1/3ln(5x)x^3-1/9x^3+C

color(white)(I)=1/9x^3(3ln(5x)-1)+C