Rules of Integration

There are basic rules of integration to follow when calculating the integral of a function. Rules of integration will make finding integrals easier. We have used the rules of integration in our examples without explicitly defined the rules of integration. Rules of integration is key to integration help for students of integral calculus.

Linear properties - Rules of integration

First rule of integration

The integral from A to B or a function f(x) and a function g(x) dx is the sum of the integral from A to B of the function f(x) dx and the integral from A to B of the function g(x) dx. In another word, the integral of the sum of two functions is the sum of the respective integrals.

Illustration of the linear properties in the rules of integration:

Let Int[A,B][f + g] be the integral from A to B of f(x) + g(x), the sum of the two functions

Let Int[A,B][f] be the integral from A to B of only the function f(x), and similarly, Int[A,B][g] be the integral from A to B of only the function g(x). Then:

Int[A,B][f + g] = Int[A,B][f] + Int[A,B][g]

Second rule of integration

If f(x) is a continuous function on [A,B] and c is a constant, then:

The integral from A to B of the function cf(x) dx is equal to c x the integral from A to B of the function f(x) dx. In another word,

Int[A,B][cf] = c Int[A,B][f]

The linear properties of integration allow any constant to be pulled out of the integration operation. 

Additivity Properties - Rules of Integration

Other rules of integration concern additivity properties on integration.

Third rule of integration

If the function f(x) is continuous on [A,B] and that A<B<C then:

The sum of the integral of f(x) dx from A to B and the integral of f(x) dx from B to C is the same as the integral from A to C of f(x) dx. Another way of illustrating this third rule of integration is:

Int[A,B][f] + Int[B,C][f] = Int[A,C][f]

Note that if f is not continuous, then this rule of integration will not hold true.