Question

please help (INTEGRATION BY PARTS)

∫x^-2 ln(x) dv

please show what u,du,dv,and v are equal to

Answer 1

\[\Large\rm \int\limits x^{-2}\ln x~dx\]So when you see a natural log, one thing to think about is "Oh yah, the derivative of ln is a power of x... so maybe I can let that be my u since it will combine nicely with the other term after I differentiate it."

I think that's what we'll want in this case.

Answer 2

\[\Large\rm u=\ln x,\qquad\qquad dv=x^{-2}dx\]

Answer 3

\[\Large\rm du=\frac{1}{x}dx,\qquad\quad v=-x^{-1}\]So we get something like that for our parts, yes?
Make sense so far?

Answer 4

yes cause we take the derivative of u and the antiderivative of dv right

Answer 5

yes.

Answer 6

I dunno why I did the different shapes underlines lol.
But whatever, I did.