H: Integration: Integrate using partial fractions that are linear in the denominator.

The idea

This topic teaches you to integrate rational expressions by first splitting them into partial fractions with linear denominators. For example, to integrate (2x+3)/((x+1)(x-2)), you write it as A/(x+1) + B/(x-2), solve for A and B, then integrate each term as A ln|x+1| + B ln|x-2| + C. The method requires the degree of the numerator to be less than the denominator; if not, perform polynomial division first. You must also handle repeated linear factors, such as (x+1)^2, by including terms A/(x+1) + B/(x+1)^2. The skill is assessed in problems that require algebraic manipulation, correct integration of 1/(ax+b), and appropriate use of modulus signs and the constant of integration.

What earns the marks

• Decompose the rational function into partial fractions with linear denominators, including cases with repeated factors.
• Integrate each partial fraction term correctly, using the standard result ∫ 1/(ax+b) dx = (1/a) ln|ax+b| + C.
• Include the constant of integration and use modulus signs inside logarithms where appropriate.
• If the numerator degree is not less than the denominator, perform polynomial division before decomposing.
• Combine logarithmic terms correctly, applying logarithm laws where simplification is required.

Common mistakes and how to put them right

• Forgetting to divide by the coefficient of x when integrating 1/(ax+b).

  Put it right: Write (1/a) ln|ax+b|.

• Omitting the modulus signs in logarithmic terms.

  Put it right: Use ln|ax+b| because the argument of a logarithm must be positive.

• Attempting partial fractions when the numerator degree is greater than or equal to the denominator.

  Put it right: First divide the polynomials to obtain a proper fraction.

• Incorrectly setting up partial fractions for repeated linear factors, e.g., using only A/(x+1) for (x+1)^2.

  Put it right: Include both A/(x+1) and B/(x+1)^2.

Exam tips

• Always check whether the rational function is proper; if not, perform polynomial division first.
• When solving for constants, substitute convenient values of x that make denominators zero to simplify equations.
• After integrating, simplify logarithmic expressions using log laws if the question asks for a single logarithm.
• Remember to include the constant of integration in indefinite integrals; it is often required for full marks.