Definite Integral Notes for Class 12 & IIT JEE
Definite Integrals evaluate the exact area under a curve over a given interval [a, b], yielding a numerical value using the Fundamental Theorem of Calculus: ∫ab f(x) dx = F(b) − F(a), and are essential for solving area, symmetry, and JEE Main calculus problems.
Table of Contents
eSaral › JEE › JEE Main ›Definite Integral Notes
🚀 Checkout eSaral Courses
What Is a Definite Integral?
A definite integral computes the exact area between a function f(x) and the x-axis over a closed interval [a, b]. Unlike an indefinite integral, it produces a specific numerical value rather than a function plus a constant.
Notation:
$$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$$
where F(x) is any antiderivative of f(x), and a and b are the lower and upper limits of integration respectively.
Definite Integral Notes
Frequently Asked Questions
Find answers to common questions.
What is a definite integral in simple terms?
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.
How many questions come from definite integrals in JEE Mains?
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.
What is the King's Property of definite integrals?
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.
Is definite integration important for JEE Advanced?
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.
What is the difference between definite and indefinite integrals?
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.