Exercise 3.2 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.2, Unit 3: Integration, for Class 12 Punjab Board students. Covers the method of substitution (change of variable), antidifferentiation of composite functions, and trigonometric substitutions for radical algebraic forms, worked out question by question.
Exercise Overview
Exercise 3.2 of Class 12 Mathematics introduces the Method of Substitution (Change of Variable), the primary technique for transforming non-standard integrands into elementary forms under the Punjab Board PCTB / PECTAA curriculum. By setting (u = g(x)) and computing the differential (du = g'(x) dx), composite expressions of the form (∫ f(g(x))·g'(x) dx) simplify directly into (∫ f(u) du).
The exercise trains students on two essential shotcut theorems derived from substitution: the Extended Power Rule (∫ [f(x)]ⁿ f'(x) dx = [f(x)]^(n+1)/(n+1) + c (for n ≠ -1)) and the Logarithmic Derivative Rule (∫ f'(x)/f(x) dx = ln|f(x)| + c). Furthermore, it covers Trigonometric Substitution for radical algebraic forms: substituting (x = a sin θ) for (√(a² - x²)), (x = a tan θ) for (√(a² + x²)), and (x = a sec(θ)) for (√(x² - a²)), leading to classical inverse trigonometric antiderivative formulas.
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 3.2 is heavily tested in Section B Short Questions and Section C Long Question parts. These step-by-step solutions highlight explicit substitution definitions, differential balancing, and back-substitution steps.