Exercise 3.3 Class 12 Math Solutions - Unit 3 Integration, Punjab Board
Step-by-step solutions to Exercise 3.3, Unit 3: Integration, for Class 12 Punjab Board students. Covers Integration by Parts, the product rule in integral form, the ILATE selection rule, cyclic integration, and the special exponential formula, worked out question by question.
Exercise Overview
Exercise 3.3 of Class 12 Mathematics explores Integration by Parts, the integral counterpart of the product rule of differentiation under the Punjab Board PCTB / PECTAA curriculum. Originating from (d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)), integration by parts provides the formula (∫ u(x)v'(x) dx = u(x)v(x) - ∫ u'(x)v(x) dx), enabling students to integrate products of dissimilar functions.
The exercise trains students on the ILATE mnemonic rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential) to systematically choose the first function (u(x)) (which differentiates into a simpler term) and the second function (v'(x)) (which integrates readily). It covers single transcendental functions integrated by inserting a dummy factor of 1 (such as (∫ ln x dx) and (∫ arcsin x dx)), repeated integration by parts, cyclic / recurring integrals where algebraic isolation of (I) is required (e.g., (∫ e^(ax) sin(bx) dx)), and the high-yield theorem (∫ e^(ax)[a f(x) + f'(x)] dx = e^(ax) f(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 3.3 is essential for Section C Long Questions and conceptual MCQs. These step-by-step solutions provide explicit (u) and (dv) assignments, clean intermediate integration steps, and clear algebraic proofs.