Chapter 1: Direct Evaluation via FTC The most straightforward approach. Find antiderivative F(x), evaluate F(b) - F(a). All indefinite integration techniques (substitution, by parts, partial fractions, trig identities) apply here. The only difference: when substituting in a definite integral, change the limits to match the new variable.

Chapter 2: King's Rule and Symmetry The dominant strategy for JEE. When you see integral(0 to a) of something involving f$\frac{x}{(f(x)}$+f(a-x)), immediately apply King's Rule. Common disguises: sin^$\frac{n}{sin^n+cos^n}$, sqrt(f)/[sqrt(f)+sqrt(g)], log variants. Also look for: integral(0,pi) x*g(sinx) which reduces to $\frac{pi}{2}$*integral g(sinx).

Chapter 3: Even/Odd Function Properties On symmetric intervals [-a,a]: check parity. Products: eveneven=even, oddodd=even, even*odd=odd. Split compound functions: $e^x$ = cosh(x) + sinh(x) where cosh is even and sinh is odd.

Chapter 4: Periodicity and Absolute Value For periodic integrands, count the number of periods. For |f(x)|, the period may be different from f(x) — |sin x| has period pi, not 2pi. For integrals of |f(x)|, split at zeros of f(x) within each period.

Chapter 5: Leibniz Rule Applications Two main JEE patterns: (1) Given F(x) = integral(0 to h(x)) f(t)dt, find F'(x). (2) Given integral(0 to x) f(t)dt = expression in x, differentiate to find f(x). The chain rule applies when limits are functions of x.

Chapter 6: Limit of Sum (Riemann Sum) Convert: replace r/n by x, 1/n by dx, sum by integral, and limits by evaluating r/n at endpoints. Generalization: lim$\frac{1}{n}$sum f$\frac{r}{n}$ from r=p to r=q gives integral from lim$\frac{p}{n}$ to lim$\frac{q}{n}$ of f(x)dx.