using integration by parts for int [(ln(3x))^2] dx

[jmf5x]

I have to integrate the following using integration by parts:

. . .int [(ln(3x))^2] dx

I just need help figuring how what to use! Please help!

 

[galactus]

Let \(\displaystyle \L\\u=(ln(3x))^{2};\;\ dv=dx;\;\ du=\frac{2ln(3x)}{x}dx;\;\ v=x\)

\(\displaystyle \L\\xln(3x)^{2}-\int{2ln(3x)}dx\)

Use parts again:

Let \(\displaystyle \L\\u=ln(3x);\;\ dv=dx;\;\ du=\frac{1}{x}dx;\;\ v=x\)

\(\displaystyle \L\\xln(3x)^{2}-2\left[xln(3x)-x\int\frac{1}{x}dx\right]\)

\(\displaystyle \L\\xln(3x)^{2}-2xln(3x)+2x\)