Definite Integration

Concept overview

The definite integral ∫_a^b f(x) dx equals F(b) − F(a) where F is any antiderivative of f (fundamental theorem of calculus). Alternatively, it's the limit of a Riemann sum — the signed area between the curve and the x-axis over [a, b].

Standard properties are the JEE workhorse. ∫_a^a f = 0. ∫_a^b f = −∫_b^a f. Linearity: ∫(af + bg) = a∫f + b∫g. ∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx (king rule). Even functions: ∫_{-a}^a f = 2∫_0^a f; odd functions: ∫_{-a}^a f = 0.

Periodic functions: for f with period T, ∫_0^T f = ∫_c^(c+T) f for any c. Also ∫_0^(nT) f = n × ∫_0^T f.

Newton-Leibniz differentiation: d/dx ∫_{g(x)}^{h(x)} f(t) dt = f(h(x)) h'(x) − f(g(x)) g'(x). This is Leibniz's rule for variable-limit integrals.

Key formulas & concepts

• Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a)
• King rule: ∫_a^b f(x) dx = ∫_a^b f(a + b − x) dx
• Odd function: ∫_{-a}^a f dx = 0 if f is odd
• Even function: ∫_{-a}^a f dx = 2∫_0^a f dx if f is even
• Periodic: ∫_0^{nT} f dx = n × ∫_0^T f dx
• Leibniz variable limits: d/dx ∫_{g(x)}^{h(x)} f dt = f(h) h' − f(g) g'

Common mistakes to avoid

• Applying the fundamental theorem to a discontinuous integrand — the antiderivative might not span the interval.
• Confusing odd/even function properties — always verify f(−x) = −f(x) or f(−x) = f(x) before applying.
• Forgetting the negative sign when swapping limits.
• Using Leibniz's variable-limit rule without accounting for chain-rule factors on the limits.

JEE Main relevance

JEE Main asks 2 definite-integration problems every paper — usually one application of properties (king, odd/even, periodic) and one evaluation.

JEE Advanced relevance

Advanced problems combine multiple properties or require creative substitution to make king rule work. Estimation of an integral by comparison with a known integral is another Advanced pattern.

Previous-year questions (PYQ)

King rule and property-based evaluation problems are the highest-frequency PYQ archetype.