Indefinite Integration

Concept overview

An indefinite integral (antiderivative) of f is any F such that F' = f. The general antiderivative is F(x) + C for arbitrary constant C. Standard antiderivatives — xⁿ, eˣ, sin, cos, 1/x — should be memorised.

Substitution u = g(x): ∫f(g(x)) g'(x) dx = ∫f(u) du. Recognising when a substitution reveals a standard form is the primary skill.

Integration by parts: ∫u dv = uv − ∫v du, chosen so the new integral is easier. The LIATE priority (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) usually picks the right u.

Partial fractions decompose a rational function P(x)/Q(x) into a sum of simpler fractions integrable term by term. When Q factors into distinct linear terms, each root gives a term A/(x − r). Repeated roots give higher powers; irreducible quadratics give (Ax + B)/(x² + bx + c) terms.

Key formulas & concepts

• Power rule: ∫xⁿ dx = x^(n+1)/(n+1) + C, n ≠ −1
• 1/x rule: ∫(1/x) dx = ln|x| + C
• Substitution: ∫f(g(x))g'(x) dx = ∫f(u) du
• By parts: ∫u dv = uv − ∫v du
• LIATE priority: L > I > A > T > E for choosing u in by-parts
• ∫eˣ dx: = eˣ + C
• ∫sec² x dx: = tan x + C

Common mistakes to avoid

• Forgetting the arbitrary constant C — indefinite integrals include it.
• Applying the power rule to n = −1 — that case gives ln|x|, not x⁰/0.
• Choosing the wrong u in by-parts — LIATE almost always gives the right choice.
• Missing factors when substituting — dx must be converted to du/g'(x).

JEE Main relevance

JEE Main asks 2 integration problems every paper — usually one substitution and one by-parts or partial-fractions.

JEE Advanced relevance

Advanced problems require multiple techniques in sequence (substitute, then by-parts) or use identities from other chapters (trig identities to transform an integrand).

Previous-year questions (PYQ)

Standard-form recognition is the top PYQ skill. Drill 30 problems across substitution, by-parts and partial fractions in three separate sessions.