Chapter 7 — Integrals

Chapter 7 of the Class 12 Maths NCERT textbook, "Integrals", covers integration as 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.

• Integration as reverse differentiation — The chapter frames integration as undoing a derivative: finding a function whose slope is the given one. Because many functions share the same slope, the indefinite integral always carries an arbitrary constant, and standard formulae come straight from known derivative rules.
• Three core integration techniques — Most integrals need reshaping first. Substitution swaps in a new variable to simplify the integrand, partial fractions break rational functions into easy pieces, and integration by parts handles products of functions — together covering the bulk of integrals students meet.
• Definite integrals and their power — Moving from families of antiderivatives to exact numbers, the definite integral is tied to indefinite integration by the Fundamental Theorem of Calculus, turning it into a practical tool for computing areas and solving problems in physics, economics and probability.

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.