Linearity of integration

A property of the integral which follows from the sum rule in integration and the constant factor rule in integration.

Let f and g be functions. Now consider:

∫ a f(x) + b g(x) dx

By the sum rule in integration, this is:

∫ (a f(x) dx) + ∫ (b g(x) dx)

By the constant factor rule in integration, this reduces to:

a ∫ f(x) dx + b ∫ g(x) dx

Hence we have:

∫ a f(x) + b g(x) dx = a ∫ f(x) dx + b ∫ g(x) dx