Question

How to integrate ∫27x/(72-x)dx ?

Answer 1

If you do long divison \[\frac{ 27x }{ 72-x }\]
becomes \[-\frac{ 1994 }{ x-72 } - 27\]
Which you can integrate as seperate functions. \[-\int\limits_{}^{}\frac{ 1994 }{ x-72 }dx -\int\limits_{}^{}27dx\]
You can then use u-substition (u=x-72, du=dx) on the first one and get\[-1994\ln \left| x-72 \right|+C\] and the second one just becomes \[-27x+C\]

Together, your answer is \[-27x-1994\ln \left| x-72 \right|+C\]