Definite & Indefinite Integrals - How It Appears in JEE

6 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Pattern

Evaluate integral using appropriate substitution

How to recognize

Composite functions, expressions like f(g(x))·g'(x), trig substitutions needed

→ Identify the inner function. Let t = g(x). Convert everything to t. For definite integrals, change limits too.

Pattern

Use King's property, even/odd, or periodicity to evaluate

How to recognize

Symmetric limits, f(a+b-x) appears naturally, trig functions with limits 0 to π or -a to a

→ Try King's property first: replace x with (a+b-x). Add to original. If limits are symmetric, check parity.

Pattern

Integral of a product requiring by-parts technique

How to recognize

Product of different types: x·eˣ, x·sinx, ln(x)·xⁿ, inverse trig · polynomial

→ Choose u by ILATE. Apply ∫u dv = uv - ∫v du. May need repeated application or tabular method.

Pattern

Convert a limit of sum to a definite integral or vice versa

How to recognize

lim(1/n)Σf(r/n), or sum with n→∞

→ Replace r/n → x, 1/n → dx. Determine limits from r range. Evaluate the resulting integral.

Pattern

Evaluate integrals involving powers of trig functions

How to recognize

sin^n(x), cos^n(x), tan^n(x), or products like sin^m·cos^n

→ Use reduction formulas, Walli's formula (for 0 to π/2), or convert using half-angle/double-angle identities.

Pattern

Given F(x) defined via integral, find values or properties

How to recognize

F(x) = ∫f(t)dt with variable limit, or differential equation involving integrals

→ Differentiate using Leibniz rule: d/dx ∫₀ˣ f(t)dt = f(x). Find f(x) first, then evaluate.