Definite Integral Notes for Class 12 & IIT JEE

A definite integral measures the exact area under a curve f(x) between two fixed points a and b on the x-axis. It gives a specific number (not a function). You compute it by finding the antiderivative F(x) and calculating F(b) − F(a). This is directly derived from the Fundamental Theorem of Calculus.

JEE Mains typically includes 1 to 2 questions from definite integrals in most sessions, as per NTA's official released papers from 2019–2024. Combined with the Area Under Curves topic, integration as a whole contributes 3–4 marks per paper — making it one of the highest-value chapters in JEE Maths.

The King's Property states that ∫[a to b] f(x) dx = ∫[a to b] f(a+b−x) dx. It is the most frequently used property in JEE because it allows the numerator and denominator of complex fractions involving sin/cos to be swapped, reducing the integral to a trivial form. It is listed as Property P4 in standard notes.

Yes. JEE Advanced tests definite integration at a deeper level than JEE Mains, focusing on the Fundamental Theorem, Leibniz's Rule for differentiation under the integral sign, and evaluating limits expressed as definite integrals (Riemann sums). At least one proof-based or multi-step integration problem appears in JEE Advanced almost every year.

An indefinite integral gives a family of functions (antiderivatives) plus an arbitrary constant C: ∫f(x)dx = F(x) + C. A definite integral evaluates this antiderivative between specific limits and returns a number: ∫[a to b] f(x)dx = F(b) − F(a). The constant C cancels out in definite integrals, which is why it is not written.