Integrating Rational Functions of the Form (ax+b)/(cx+d)

Task goals

• Foundation: Rewrite a simple rational function (ax+b)/(cx+d) using division to isolate a constant term plus a fraction with constant numerator, then integrate directly to obtain a linear term and a $\ln|cx+d|$ term.
• Proficient: Integrate rational functions of this form over a stated interval to evaluate a definite integral, including correct handling of the modulus in the log term and simplifying the constant of integration.
• Excellence: Apply this integration technique within a multi-step problem, such as finding an area between a rational function and an axis or combining it with another integration rule, and justify each algebraic step.