Chapter 1: Substitution Method The most fundamental technique. Look for a composite function f(g(x)) multiplied by g'(x). Set u = g(x), du = g'(x) dx, and the integral transforms to integral f(u) du. Common recognizable patterns: integral f(ax+b) dx = F(ax+b)/a + C (linear substitution), integral f(x^n)*x^(n-1) dx (power substitution), and integral f(trig(x))*trig'(x) dx (trig substitution).

Chapter 2: Integration by Parts Used when the integrand is a product of two different types of functions. The LIATE rule determines u: Logarithmic functions get highest priority, then Inverse trig, Algebraic, Trigonometric, and Exponential (lowest priority). Critical special cases: integral ln x dx = x ln x - x + C, integral x^n e^x dx (tabular method efficient), integral e^x sin x or e^x cos x (cyclic by-parts, solve algebraically for I).

Chapter 3: Partial Fractions For rational functions P(x)/Q(x) with deg P < deg Q. Decomposition rules by factor type of Q(x): (1) Linear factor (ax+b) gives A/(ax+b), (2) Repeated linear (ax+b)^n gives sum of A_k/(ax+b)^k for k=1 to n, (3) Irreducible quadratic (ax^2+bx+c) gives (Ax+B)/(ax^2+bx+c). Always perform long division first if deg P >= deg Q. Cover-up method provides quick solutions for distinct linear factors.

Chapter 4: Trigonometric Integrals Strategy depends on the form. For sin^m x cos^n x: if m or n is odd, save one factor for du and convert rest using sin^2+cos^2=1. If both are even, use half-angle formulas. For sin(mx)cos(nx), sin(mx)sin(nx), cos(mx)cos(nx), use product-to-sum identities. For powers of tan and sec: integral tan^n x uses the recurrence I_n = tan^(n-1)x/(n-1) - I_(n-2).

Chapter 5: Special Techniques The e^x[f(x)+f'(x)] pattern appears frequently in JEE: always check if the integrand can be decomposed this way. Completing the square converts ax^2+bx+c forms to standard (u^2+k^2) or (u^2-k^2) types. The Weierstrass substitution t = tan(x/2) is the universal method for rational trig integrals but should be a last resort due to algebraic complexity.