Chapter 7 — Integrals

NCERT Class 12 Maths Part II Chapter 7 covers Integrals — the inverse process of differentiation — including indefinite integrals, standard formulae, and methods such as substitution, partial fractions, and integration by parts, as well as definite integrals and the Fundamental Theorem of Calculus.

Chapter 7 of NCERT Class 12 Mathematics Part II introduces Integral Calculus as the inverse of differential calculus. The chapter defines the indefinite integral ∫f(x)dx = F(x) + C, establishes standard formulae derived from known derivatives, and develops three main techniques: integration by substitution (changing the variable to simplify the integrand), integration using partial fractions (decomposing rational functions into simpler terms), and integration by parts (handling products of functions via the rule ∫u dv = uv − ∫v du). It concludes with definite integrals, the Fundamental Theorem of Calculus, and applications to area and other problems in science and engineering.

1. 01Integration is the inverse process of differentiation; the anti-derivative F(x) of f(x) satisfies F′(x) = f(x), and the general indefinite integral is written as F(x) + C where C is an arbitrary constant.
2. 02Standard integral formulae are derived directly from differentiation rules — for example, ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ −1), ∫cos x dx = sin x + C, and ∫(1/x) dx = log|x| + C.
3. 03Integration by substitution transforms ∫f(x)dx into ∫f(g(t))g′(t)dt by letting x = g(t); key results derived this way include ∫tan x dx = log|sec x| + C and ∫sec x dx = log|sec x + tan x| + C.
4. 04Integration by partial fractions decomposes a proper rational function P(x)/Q(x) into a sum of simpler fractions whose integrals are standard, covering five canonical forms including linear, repeated linear, and irreducible quadratic factors.
5. 05Integration by parts uses the formula ∫f(x)g(x)dx = f(x)∫g(x)dx − ∫[f′(x)∫g(x)dx]dx; a special case gives ∫e^x[f(x) + f′(x)]dx = e^x f(x) + C.
6. 06The Fundamental Theorem of Calculus connects indefinite and definite integrals, making the definite integral a practical tool for computing areas and solving problems in economics, physics, and probability.