Integrating expressions containing inverse functions

The problem of integrating simple expressions containing inverse functions relies on the well-known method of integration by parts. For example, consider the integral
\[\int \ln x \, d x.\]
Since the natural logarithmic function is the inverse of the exponential function, the standard approach proceeds by recognising that the unit function $f(x) = 1$ can always be written as a product with the inverse function before the method of integration by parts is employed. In the case of the example given above, integration by parts leads to the well-known result of
\[\int \ln x \, dx = \int 1 \cdot \ln x \, dx = x \ln x - x +C.\]
For convenience, in the remainder of the paper we will drop the arbitrary constant of integration appearing in all indefinite integrals.