Definite Integration is one of the highest-weighted topics in JEE Main mathematics, with 3-4 questions per year. It combines the mechanical skill of integration with the strategic use of properties to simplify integrals that would otherwise be extremely difficult.
The Fundamental Theorem of Calculus (FTC) is the bridge: if F'(x) = f(x), then integral(a to b) f(x) dx = F(b) - F(a). This eliminates the constant of integration and gives a definite numerical value. FTC Part 1 says G(x) = integral(a to x) f(t) dt satisfies G'(x) = f(x), connecting differentiation and integration as inverse operations.
The properties of definite integrals are the key competitive advantage. King's Rule (integral(a,b) f(x) dx = integral(a,b) f(a+b-x) dx) is the single most important property. Its most powerful application: when f+f(a-x)) appears as integrand on [0,a], adding I + I gives 2I = a, so I = a/2. This pattern appears in nearly every JEE paper with variations involving sin/cos, sqrt, log, and powers.
The even/odd properties on symmetric intervals [-a,a] provide instant simplification: even functions double, odd functions vanish. The periodicity property reduces integrals over multiple periods to a single period.
Leibniz Rule handles differentiation under the integral sign: d/dx integral(g(x) to h(x)) f(t) dt = f(h(x))*h'(x) - f(g(x))*g'(x). JEE uses this to create functional equations where integral(0 to x) f(t) dt = some expression, and differentiating reveals f(x).
The Riemann sum connection converts limit-of-sum problems to definite integrals: lim sum f = integral(0 to 1) f(x) dx. This appears frequently for computing limits involving n-term sums.
Wallis' formula provides integral x dx as a product of decreasing fractions, with a pi/2 factor for even n. Combined with Queen's Rule and periodicity, it handles all trigonometric power integrals.
Estimation techniques using bounding (m(b-a) <= integral <= M(b-a)) and comparison (f >= g implies integral f >= integral g) appear in assertion-reasoning and multiple-correct questions.