Exercise 3.4 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.4, Unit 3: Integration, for Class 12 Punjab Board students. Covers the technique of integrating rational algebraic functions using partial fraction decomposition across linear, repeated, and irreducible quadratic denominator factors, worked out question by question.
Exercise Overview
Exercise 3.4 of Class 12 Mathematics is dedicated to the Integration of Rational Algebraic Functions using Partial Fraction Decomposition under the Punjab Board PCTB / PECTAA curriculum. A rational function (P(x) / Q(x)) represents the ratio of two polynomials. When the degree of (P(x)) is greater than or equal to the degree of (Q(x)) (improper fraction), students first execute polynomial long division to obtain a polynomial quotient plus a proper rational remainder.
The exercise systematically covers all algebraic denominator factorization cases: non-repeated linear factors (A/(x - a) + B/(x - b)), repeated linear factors (A/(x - a) + B/(x - a)²), and irreducible quadratic factors ((Ax + B) / (ax² + bx + c)). After evaluating the unknown partial fraction coefficients through zero-factor substitution and equating coefficients of identical powers of (x), students integrate the decomposed fractions into standard logarithmic forms (ln|x - a|), negative power terms, and inverse tangent integrals ((1/a) arctan(x/a)).
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 3.4 is one of the most frequently asked sections for Section C Long Questions. These step-by-step solutions present complete division steps, explicit constant determinations, and tidy integral evaluations for every problem.