∫xⁿ dx = xⁿ⁺¹/(n+1) for n ≠ −1; ∫sin x dx = −cos x; ∫cos x dx = sin x; ∫eˣ dx = eˣ; ∫dx = x.

• Integration is the reverse process of differentiation: if d/dx[F(x)] = f(x), then ∫f(x) dx = F(x) + C.
• The power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) fails at n = −1; for that case, use ∫1/x dx = ln|x| + C.
• Common NEET trap: forgetting the constant of integration C in indefinite integrals — NEET options may include answers that differ only by the presence or absence of C.

Substitution: set u = inner function, replace dx with du/g′(x). By parts: ∫uv dx = u∫v dx − ∫(du/dx × ∫v dx) dx.

• Integration by substitution converts a composite integrand into a standard form — always identify the 'inner function' as the substitution variable.
• Integration by parts is used when the integrand is a product of two functions: choose u as the function that simplifies on differentiation (LIATE rule).
• Common NEET trap: applying by-parts when simple substitution suffices, wasting time — always check if the integrand fits a standard formula before resorting to by-parts.

∫(a to b) f(x) dx = F(b) − F(a), where a is the lower limit and b is the upper limit. Geometrically, this equals the area under the curve f(x) between x = a and x = b.

• Always evaluate F(b) first, then subtract F(a) — reversing the order flips the sign of the result.
• In physics, definite integrals give scalar quantities: work (W = ∫F dx), impulse (J = ∫F dt), and displacement (s = ∫v dt) over specified intervals.
• Common NEET trap: in the Coulomb integral ∫(r₁ to r₂) 1/r² dr = [−1/r] from r₁ to r₂, forgetting the negative sign from integrating r⁻² leads to an inverted potential-energy expression.