Integrals — Undoing Derivatives

Integration is the reverse of differentiation.

Indefinite integral: ∫f(x)dx = F(x) + C, where F'(x) = f(x).

The +C is the constant of integration. NEVER forget it on indefinite integrals.

Basic rules:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

∫1/x dx = ln|x| + C

∫eˣ dx = eˣ + C

∫sin x dx = -cos x + C

∫cos x dx = sin x + C

∫sec²x dx = tan x + C

Key insight: Every derivative rule has a corresponding integral rule. If d/dx[sin x] = cos x, then ∫cos x dx = sin x + C.

AP trap: Forgetting +C on indefinite integrals costs points every year.

Reference:

Wikipedia: Integral

https://en.wikipedia.org/wiki/Integral