Integration

If f(x) and g(x) are functions

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). Written as ∫g(x)dx = f(x) + c,

Key formulas include

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. For functions of the form (ax+b), we

When dealing with composite functions

When dealing with composite functions, we use substitution: if x = φ(t), then ∫f(x)dx = ∫f[φ(t)]φ'(t)dt. Key patterns include: ∫f(x)·f'(x)dx leads to

For products of functions

For products of functions, use ∫u·v dx = u∫v dx - ∫(du/dx·∫v dx)dx. Choose the first function using 'LIAE' priority: Logarithmic, Inverse trigonometri

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

For rational functions P(x)/Q(x) where degree of P < degree of Q, decompose into simpler fractions. For example: (x+1)/[(x-2)(x+3)] = A/(x-2) + B/(x+3