Integrals

If F'(x) = f(x)

If F'(x) = f(x), then F(x) is called an anti-derivative of f(x). The indefinite integral ∫f(x)dx = F(x) + C represents the entire family of anti-deriv

Key formulas include

Key formulas include: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n≠-1), ∫1/x dx = log|x| + C, ∫eˣdx = eˣ + C, ∫cos x dx = sin x + C, ∫sin x dx = -cos x + C, ∫sec²x dx =

Linear properties

Linear properties: ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx and ∫kf(x)dx = k∫f(x)dx. The differentiation and integration are inverse processes: d/dx[∫f(x)

When direct integration is difficult

When direct integration is difficult, we substitute x = g(t) to transform ∫f(x)dx into ∫f(g(t))g'(t)dt. The key is choosing a substitution where the d

For rational functions P(x)/Q(x) where degree

For rational functions P(x)/Q(x) where degree of P(x) < degree of Q(x), we decompose into simpler fractions. Different forms exist based on the factor