Integration — Formula Sheet

Linear combination rule

∫[af(x) + bg(x)]dx = a∫f(x)dx + b∫g(x)dx

Integration of expressions involving √(x² ± a²)

Formula

∫u·v dx = u∫v dx - ∫(du/dx·∫v dx)dx

If f(x) and g(x) are functions such that d/dx[f(x)] = g(x), then f(x) is called the integral or antiderivative of g(x).

Key formulas include

∫xⁿdx = xⁿ⁺¹/(n+1) + c (n≠-1), ∫(1/x)dx = log|x| + c, ∫eˣdx = eˣ + c, ∫aˣdx = aˣ/log a + c.

When dealing with composite functions, we use substitution

if x = φ(t), then ∫f(x)dx = ∫f[φ(t)]φ'(t)dt.