Exercise 3.1 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.1, Unit 3: Integration, for Class 12 Punjab Board students. Covers antiderivatives, indefinite integrals, fundamental integration rules (power rule, constant multiple, sum and difference), and standard integration formulas, worked out question by question.
Exercise Overview
Exercise 3.1 marks the beginning of Unit 3: Integration in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum. Integration is introduced as the reverse process of differentiation (antidifferentiation). An antiderivative of a function (f(x)) is any function (F(x)) such that (F'(x) = f(x)), leading to the definition of the indefinite integral (∫ f(x) dx = F(x) + c), where (c) is the arbitrary constant of integration representing a family of parallel curves.
The exercise systematically builds foundational operational rules of integration: the Constant Rule (∫ k dx = kx + c), the Constant Multiple Rule (∫ k f(x) dx = k ∫ f(x) dx), the Sum and Difference Rule (∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx), and the fundamental Power Rule of Integration (∫ xⁿ dx = x^(n+1)/(n+1) + c (for n ≠ -1)). Students also learn to integrate reciprocal terms (∫ (1/x) dx = ln|x| + c), basic exponentials (∫ e^(ax) dx = (1/a) e^(ax) + c), and fundamental trigonometric terms such as (∫ sin x dx = -cos x + c) and (∫ sec² x dx = tan x + c).
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 3.1 serves as the prerequisite foundation for the entire integral calculus section. It frequently provides essential short questions and formula-based MCQs. These step-by-step solutions provide rigorous algebraic manipulation and clearly show every integration step.