Integration and its Applications — Formula Sheet

If d/dx[F(x)] = f(x), then ∫f(x)dx = F(x) + C

Rule

∫f(g(x))g'(x)dx = ∫f(t)dt, where g(x) = t

Formula

∫uv' dx = uv - ∫u'v dx

If d(F(x))/dx = f(x), then ∫f(x)dx = F(x) + C, where C is an arbitrary constant.

Key formulae include

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

Rule

∫f(g(x))g'(x)dx = ∫f(t)dt where g(x) = t.

For example, 1/((x-1)(x+3)) = A/(x-1) + B/(x+3).

Used for integrating products of functions using the formula

∫f(x)g(x)dx = f(x)∫g(x)dx - ∫[f'(x)∫g(x)dx]dx.