Question:

$\int \underset{II}{x^4} \cdot \underset{I}{\ln x} dx = \ln x \int x^4 dx - \int \left( \frac{d(\ln x)}{dx} \int x^4 dx \right) dx = \ln x \cdot \frac{x^5}{5} - \int \frac{1}{x} \cdot \frac{x^5}{5} dx = \frac{x^5}{5} \ln x - \frac{x^5}{25} + C$