Advanced integration techniques extend beyond basic substitution and parts to handle rational, irrational, and trigonometric integrands that arise in JEE problems. Mastery of these methods ensures efficiency in solving definite integration and differential equation problems.

Partial Fractions: Every proper rational function P(x)/Q(x) decomposes into simpler fractions based on the factorization of Q(x). Distinct linear factors give A/(x-a) terms, repeated linear factors (x-a)^k give k separate terms A1/(x-a) + ... + Ak/(x-a)^k, and irreducible quadratic factors give (Ax+B)/(x^2+px+q) terms. The cover-up method quickly finds coefficients for distinct linear factors. For improper fractions, long division must precede decomposition.

Trigonometric Substitutions: Three standard substitutions handle square roots of quadratics: x = a sin(theta) for sqrt(a^2-x^2), x = a tan(theta) for sqrt(a^2+x^2), and x = a sec(theta) for sqrt(x^2-a^2). For general quadratic expressions, complete the square first to reduce to standard form.

Reduction Formulas: These recurrence relations express I_n in terms of I_(n-2) or I_(n-1), derived via integration by parts. The key formulas are for sin^n, cos^n (reduce by 2 with coefficient (n-1)/n), tan^n (reduce by 2 with alternating signs), sec^n (reduce by 2), x^n*e^x (reduce by 1), and (ln x)^n (reduce by 1). Base cases must be memorized: I_0 for sin^n is x, I_1 is -cos x.

Wallis' Formula: For definite integrals from 0 to pi/2, boundary terms vanish, giving the pure recurrence W_n = (n-1)/n * W_(n-2). The chain terminates at W_0 = pi/2 (even n, includes pi/2 factor) or W_1 = 1 (odd n, no pi/2).

Special Forms: The e^x[f+f'] pattern gives integral = e^x*f(x) + C. The Weierstrass substitution t = tan(x/2) converts rational trigonometric integrals to rational functions. The (px+q)/sqrt(ax^2+bx+c) form splits into a derivative term and a standard form.

Standard Integrals: Six fundamental forms involving 1/(x^2+/-a^2), 1/sqrt(x^2+/-a^2), and sqrt(x^2+/-a^2) must be memorized. These, combined with completing the square, handle most quadratic-denominator integrals.