Integration Techniques — Chapter-wise — Summary

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 $\frac{ax+b}{a}$ + 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 $\frac{x}{Q}$ (x) with deg P < deg Q. Decomposition rules by factor type of Q(x): (1) Linear factor (ax+b) gives $\frac{A}{ax+b}$ , (2) Repeated linear (ax+b)^n gives sum of $\frac{A_k}{ax+b}$ ^k for k=1 to n, (3) Irreducible quadratic ( $ax^{2+bx+c}$ ) gives $\frac{Ax+B}{(ax^2+bx+c)}$ . 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) $\frac{x}{n-1}$ - 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 $\frac{x}{2}$ is the universal method for rational trig integrals but should be a last resort due to algebraic complexity.