I = int \ sin (ln x) \ dx
this is in the IBP section meaning you don't really have much choice how to take this, so ...
I =int \ d/dx(x) * sin (ln x) \ dx
which by IBP
= x sin(ln x) - int \ x *d/dx( sin (ln x)) \ dx
= x sin(ln x) - int \ x cos (ln x)* 1/x \ dx
= x sin(ln x) - int \ cos (ln x) \ dx
and now another round of IBP
= x sin(ln x) - int \ d/dx(x) * cos (ln x) \ dx
= x sin(ln x) - x cos(ln x) + int \ x d/dx( cos (ln x)) \ dx
= x sin(ln x) - x cos(ln x) + int \ x (-sin (ln x) 1/x) \ dx
= x sin(ln x) - x cos(ln x) - int \ sin (ln x) \ dx
= x sin(ln x) - x cos(ln x) - I
2I = x sin(ln x) - x cos(ln x) +C making sure to add in the integration constant
implies I = x/2 ( sin(ln x) - cos(ln x) )+C