Exercise 3.5 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.5, Unit 3: Integration, for Class 12 Punjab Board students. Covers the Definite Integral, the Fundamental Theorem of Calculus (FTC), definite integration by parts, and key integral properties (interval splitting, symmetry, and limits), worked out question by question.
Exercise Overview
Exercise 3.5 of Class 12 Mathematics transitions from indefinite integration to the theory and evaluation of The Definite Integral under the Punjab Board PCTB / PECTAA curriculum. At the center of this transition is the Fundamental Theorem of Calculus (FTC), which connects differential and integral calculus by stating that if (F(x)) is an antiderivative of a continuous function (f(x)) on ([a, b]), then (∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) - F(a)).
The exercise trains students in applying definite integration by parts (∫ₐᵇ u(x)v'(x) dx = [u(x)v(x)]ₐᵇ - ∫ₐᵇ u'(x)v(x) dx) alongside the essential algebraic properties of definite integrals. These properties include the reversing of limits (∫ᵦᵃ f(x) dx = −∫ₐᵇ f(x) dx), interval additivity (∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫꜀ᵇ f(x) dx) (vital for piecewise and absolute value functions), symmetry properties for even functions (2∫₀ᵃ f(x) dx) and odd functions (∫₋ₐᵃ f(x) dx = 0), and the dummy variable invariance property.
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 3.5 is a frequent source of conceptual short questions and Section C Long Questions. These step-by-step solutions provide rigorous boundary evaluations, limit changes during substitution, and clear geometric justifications for each problem.