Definite Integration for JEE Main: Techniques & Properties

Definite integration is one of the most formula-rich and technique-dependent chapters in JEE Main Mathematics, contributing three to five marks every session. The chapter has two components: the standard properties of definite integrals (which allow simplification before evaluation), and the evaluation techniques themselves (substitution, integration by parts, partial fractions). Mastering the properties first creates massive computational shortcuts — many integrals that look difficult simplify to elementary calculations once the right property is applied.

Essential Properties of Definite Integrals

Property 1: ∫_a^b f(x)dx = −∫_b^a f(x)dx (reversing limits changes sign). Property 2: ∫_a^b f(x)dx = ∫_a^b f(a+b−x)dx (King's property — replace x with a+b−x). This is the most tested property in JEE; it converts many integrals into their negatives, revealing that I = −I, so I = 0, or converts a complicated integrand to a simple one. Property 3: ∫_0^{2a} f(x)dx = 2∫_0^a f(x)dx if f(2a−x) = f(x), or = 0 if f(2a−x) = −f(x). Property 4: ∫_{−a}^a f(x)dx = 2∫_0^a f(x)dx for even f(x) [f(−x) = f(x)], or = 0 for odd f(x) [f(−x) = −f(x)].

The even/odd property is the second most tested: if the integrand is odd on a symmetric interval [−a, a], the integral is immediately zero — no calculation needed. JEE presents these as "find ∫_{−π}^π x⁴·sinx dx" (odd function on symmetric interval → 0) or disguises the odd function inside a composite. The ability to spot odd/even functions instantly saves minutes per question. Take a free integration mock test to practice property recognition.

Leibniz Rule and Differentiation Under the Integral Sign

Leibniz rule: d/dx ∫_{g(x)}^{h(x)} f(t,x)dt = f(h(x),x)·h'(x) − f(g(x),x)·g'(x) + ∫_{g(x)}^{h(x)} ∂f/∂x dt. In JEE, the most common form has a variable upper limit: d/dx ∫_a^{g(x)} f(t)dt = f(g(x))·g'(x). This appears in "find the function whose integral from 0 to x satisfies a given functional equation" or in problems involving the derivative of a definite integral with variable limits.

Walli's formula for ∫_0^{π/2} sinⁿx dx = ∫_0^{π/2} cosⁿx dx: if n is even, = [(n−1)!! / n!!] × π/2; if n is odd, = [(n−1)!! / n!!], where n!! is the double factorial. JEE tests ∫_0^{π/2} sin⁶x dx or similar — Walli's formula makes this a one-minute calculation rather than a multi-step integration by parts.

Evaluation Techniques: Substitution and Integration by Parts

Substitution (change of variables): replace x with a new variable u = g(x), changing the integrand and the limits simultaneously. The definite integral limits transform with the substitution: if x=a gives u=α and x=b gives u=β, then ∫_a^b f(x)dx = ∫_α^β f(g(u))g'(u)du. Trigonometric substitution for integrals involving √(a²−x²), √(a²+x²), √(x²−a²) uses x = a sinθ, x = a tanθ, x = a secθ respectively, converting to pure trigonometric integrals.

Integration by parts: ∫u dv = uv − ∫v du. The ILATE priority rule (Inverse trig > Logarithm > Algebraic > Trigonometric > Exponential) selects which factor to differentiate (u) and which to integrate (dv). Reduction formulae — expressing ∫sinⁿx dx in terms of ∫sinⁿ⁻²x dx — arise naturally from repeated integration by parts. For the area applications of definite integration, see our area between curves guide. For the full calculus picture, see our limits and continuity guide.