find the definite integral between limits 1 and 2 of (4x^3+1)/(x^4+x) with respect to x

first notice the integral is in the form f'(x)/f(x), and indefinite integrals of this form are ln|f(x)|+c.
therefore the integral is [ln|x⁴+x|] between limits 1 and 2.
subbing in limits gives ln|2⁴+2|-ln|1⁴+1|
simplifying gives ln|18|-ln|2|
and by log rules this is equivalent to ln|18/2|=ln|9|.